Transfinite diameter, Chebyshev constant and energy on locally compact
  spaces
arXiv:0704.0859v1  [math.CA]  6 Apr 2007
Transfinite diameter, Chebyshev constant and energy
on locally compact spaces
Bálint Farkas∗ (farkas@mathematik.tu-darmstadt.de)
Technische Universität Darmstadt, Fachbereich Mathematik
Department of Applied Analysis
Schloßgartenstraße 7, D-64289, Darmstadt, Germany
Béla Nagy† (nbela@sol.cc.u-szeged.hu)
Bolyai Institute, University of Szeged
Aradi vértanúk tere 1
H-6720, Szeged, Hungary
Abstract. We study the relationship between transfinite diameter, Chebyshev
constant and Wiener energy in the abstract linear potential analytic setting pio-
neered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential
theoretic kernel has the maximum principle, then all these quantities are equal for
all compact sets. For continuous kernels even the converse statement is true: if the
Chebyshev constant of any compact set coincides with its transfinite diameter, the
kernel must satisfy the maximum principle. An abundance of examples is provided
to show the sharpness of the results.
Keywords: Transfinite diameter, Chebyshev constant, energy, potential theoretic
kernel function in the sense of Fuglede, Frostman’s maximum principle, rendezvous
and average distance numbers.
2000 Math. Subj. Class.: 31C15; 28A12, 54D45
Dedicated to the memory of Professor Gustave Choquet (1 March
1915 - 14 November 2006)
1. Introduction
The idea behind abstract (linear) potential theory, as developed by
Choquet [4], Fuglede [9] and Ohtsuka [15], is to replace the Euclidian
space Rd by some locally compact space X and the well-known Newto-
nian kernel by some other kernel function k : X×X → R∪{+∞}, and
∗ This work was started during the 3rd Summerschool on Potential Theory, 2004,
hosted by the College of Kecskemét, Faculty of Mechanical Engineering and Automa-
tion (GAMF). Both authors would like to express their gratitude for the hospitality
and the support received during their stay in Kecskemét.
† The second named author was supported by the Hungarian Scientific Research
Fund; OTKA 49448
http://arxiv.org/abs/0704.0859v1
to look at which “potential theoretic” assertions remain true in this gen-
erality (see the monograph of Landkof [12]). This approach facilitates
general understanding of certain potential theoretic phenomena and
allows also the exploration of fundamental principles like Frostman’s
maximum principle.
Although there is a vast work done considering energy integrals and
different notions of energies, the familiar notions of transfinite diame-
ter and Chebyshev constants in this abstract setting are sporadically
found, sometimes indeed inaccessible, in the literature, see Choquet [4]
or Ohtsuka [17]. In [4] Choquet defines transfinite diameter and proves
its equality with the Wiener energy in a rather general situation, which
of course covers the classical case of the logarithmic kernel on C. We
give a slightly different definition for the transfinite diameter that, for
infinite sets, turns out to be equivalent with the one of Choquet. The
primary aim of this note is to revisit the above mentioned notions and
related results and also to partly complement the theory.
We already remark here that Zaharjuta’s generalisation of transfi-
nite diameter and Chebyshev constant to Cn is completely different in
nature, see [24], whereas some elementary parts of weighted potential
theory (see, e.g., Mhaskar, Saff [13] and Saff, Totik [20]) could fit in
this framework.
The power of the abstract potential analytic tools is well illustrated
by the notion of the average distance number from metric analysis, see
Gross [11], Stadje [21]. The surprising phenomenon noticed by Gross is
the following: If (X, d) is a compact connected metric space, there al-
ways exists a unique number r(X) (called the average distance number
or the rendezvous number of X), with the property that for any finite
point system x1, . . . , xn ∈ X there is another point x ∈ X with average
distance
d(xj , x) = r(X).
Stadje generalised this to arbitrary continuous, symmetric functions
replacing d. Actually, it turned out, see the series of papers [6, 5, 7] and
the references therein, that many of the known results concerning av-
erage distance numbers (existence, uniqueness, various generalisations,
calculation techniques etc.), can be proved in a unified way using the
works of Fuglede and Ohtsuka. We mention for example that Frost-
man’s Equilibrium Theorem is to be accounted for the existence for
certain invariant measures (see Section 5 below). In these investigations
the two variable versions of Chebyshev constants and energies and even
their minimax duals had been needed, and were also partly available
due to the works of Fuglede [10] and Ohtsuka [16, 17], see also [6].
Another occurrence of abstract Chebyshev constants is in the study
of polarisation constants of normed spaces, see Anagnostopoulos, Ré-
vész [1] and Révész, Sarantopoulos [19].
Let us settle now our general framework. A kernel in the sense of
Fuglede is a lower semicontinuous function k : X × X → R ∪ {+∞}
[9, p. 149]. In this paper we will sometimes need that the kernel is
symmetric, i.e., k(x, y) = k(y, x). This is for example essential when
defining potential and Chebyshev constant, otherwise there would be
a left- and right-potential and the like.
Another assumption, however a bit of technical flavour, is the pos-
itivity of the kernel. This we need, because we would like to avoid
technicalities when integrating not necessarily positive functions. This
assumption is nevertheless not very restrictive. Since we usually con-
sider compact sets of X ×X, where by lower semicontinuity k is nec-
essarily bounded from below, we can assume that k ≥ 0. Indeed, as we
will see, energy, nth diameter and nth Chebyshev constant are linear in
constants added to k.
Denote the set of compactly supported Radon measures on X by
M(X), that is
M(X) := {µ : µ is a regular Borel measure on X,
µ has compact support, ‖µ‖ < +∞}.
Further, let M1(X) be the set of positive unit measures from M(X),
M1(X) := {µ ∈ M(X) : µ ≥ 0, µ(X) = 1}.
We say that µ ∈ M1(X) is supported on H if supp µ, which is
a compact subset of X, is in H. The set of (probability) measures
supported on H are denoted by M(H) (M1(H)).
Before recalling the relevant potential theoretic notions from [9] (see
also [15]), let us spend a few words on integrals (see [2, Ch. III-IV.]). Let
µ be a positive Radon measure on X. Then the integral of a compactly
supported continuous function with respect to µ is the usual integral.
The upper integral of a positive l.s.c. function f is defined as
f dµ := sup
0 ≤ h ≤ f
h ∈ Cc(X)
h dµ.
This definition works well, because by standard arguments (see, e.g.,
[2, Ch. IV., Lemma 1]) one has
k(x, y) = sup
0 ≤ h ≤ k
h ∈ Cc(X ×X)
h(x, y),
where, because of the symmetry assumption, it suffices to take only
symmetric functions h in the supremum.
What should be here noted, is that this notion of integral has all
useful properties that we are used to in case of Lebesgue integrals (note
also the necessity of the positivity assumptions).
The usual topology onM is the so-called vague topology which is a lo-
cally convex topology defined by the family {µ 7→
X f dµ : f ∈ Cc(X)}
of seminorms. We will only encounter this topology in connection with
families M of measures supported on subsets of the same compact set
K ⊂ X. In this case, the weak∗-topology (determined by C(K)) and
the vague topology coincide on M, Fuglede [9].
For a potential theoretic kernel k : X ×X → R+ ∪ {0} Fuglede [9]
and Ohtsuka [15] define the potential and the energy of a measure µ
Uµ(x) :=
k(x, y) dµ(y) , W (µ) :=
k(x, y) dµ(y) dµ(x).
The integrals exist in the above sense, although may attain +∞ as
well.
For a given set H ⊂ X its Wiener energy is
w(H) := inf
µ∈M1(H)
W (µ), (1)
see [9, (2) on p. 153].
One also encounters the quantities (see [9, p. 153])
U(µ) := sup
Uµ(x), V (µ) := sup
x∈ supp µ
Uµ(x).
Accordingly one defines the following energy functions
u(H) := inf
µ∈M1(H)
U(µ), v(H) := inf
µ∈M1(H)
V (µ).
In general, one has the relation
w ≤ v ≤ u ≤ +∞,
where in all places strict inequality may occur. Nevertheless, under our
assumptions we have the equality of the energies v and w, being gen-
erally different, see [9, p. 159]. More importantly, our set of conditions
suffices to have a general version of Frostman’s equilibrium theorem,
see Theorem 9.
In fact, at a certain point (in §4), we will also assume Frostman’s
maximum principle, which will trivially guarantee even u = v, that is,
the equivalence of all three energies treated by Fuglede.
Definition. The kernel k satisfies the maximum principle, if for every
measure µ ∈ M1
U(µ) = V (µ).
As our examples show in §5, this is essential also for the equivalence
of the Chebyshev constant and the transfinite diameter. Carleson [3,
Ch. III.] gives a class of examples satisfying the maximum principle:
Let Φ(r), r = |x|, x ∈ Rd be the fundamental solution of the Laplace
equation, i.e., Φ(|x−y|) the Newtonian potential on Rd. For a positive,
continuous, increasing, convex function H assume also that
H(Φ(r))rd−2 dr < +∞.
Then H ◦Φ satisfies the maximum principle; see [3, Ch. III.] and also
Fuglede [9] for further examples.
Let us now turn to the systematic treatment of the Chebyshev
constant and the transfinite diameter. We call a function g : X →
R log-polynomial, if there exist w1, . . . , wn ∈ X such that g(x) =
j=1 k(x,wj) for all x ∈ X. Accordingly, we will call the wjs and
n the zeros and the degree of g(x), respectively. Obviously the sum of
two log-polynomials is a log-polynomial again. The terminology here is
motivated by the case of the logarithmic kernel
k(x, y) = − log |x− y|,
where the log-polynomials correspond to negative logarithms of alge-
braic polynomials.
Log-polynomials give access to the definition of transfinite diameter
and the Chebyshev constant, see Carleson [3], Choquet [4], Fekete [8],
Ohtsuka [17] and Pólya, Szegő [18]. First we start with the “degree n”
versions, whose convergence will be proved later.
Definition. Let H ⊂ X be fixed. We define the nth diameter of H as
Dn(H) := inf
w1,...,wn∈H
(n− 1)n
1≤j 6=l≤n
k(wj , wl)
; (2)
or, if the kernel is symmetric
Dn(H) = inf
w1,...,wn∈H
(n− 1)n
1≤i<j≤n
k(wi, wj)
If H is compact, then due to the fact that k is l.s.c., Dn(H) is
attained for some points w1, . . . , wn ∈ H, which are then called n-Fekete
points. We will also use the term approximate n-Fekete points with the
obvious meaning. Note also that for a finite set H, #H = m and
n > m, there is always a point from the diagonal ∆ = {(x, x) : x ∈ H}
in the definition of Dn(H). This possibility is completely excluded by
Choquet in [4], thus allowing only infinite sets.
Definition. For an arbitrary H ⊂ X the nth Chebyshev constant of
H is defined as
Mn(H) := sup
w1,...,wn∈H
k(x,wk)
We are going to show that both nth diameters and nth Chebyshev
constants converge from below to some number (or +∞), which are
respectively called the transfinite diameter D(H) and the Chebyshev
constant M(H). The aim of this paper is to relate these quantities as
well as the Wiener energy of a set.
2. Chebyshev constant and transfinite diameter
We define the Chebyshev constant and the transfinite diameter of a
set H ⊂ X and proceed analogously to the classical case. It turns out,
though not very surprisingly, that in general the equality of these two
quantities does not hold.
First, we prove the convergence of nth diameters and nth Chebyshev
constants. This is for both cases classical, we give the proof only for
the sake of completeness, see, e.g., Carleson [3], Choquet [4], Fekete [8],
Ohtsuka [17] and Pólya, Szegő [18].
PROPOSITION 1. The sequence of nth diameters is monotonically
increasing.
Proof. Choose x1, . . . , xn ∈ H arbitrarily. If we leave out any index
i = 1, 2, . . . , n, then for the remaining n − 1 points we obtain by the
definition of Dn−1(H) that
(n− 1)(n − 2)
1 ≤ j 6= l ≤ n
j 6= i, l 6= i
k(xj , xl) ≥ Dn−1(H).
After summing up for i = 1, 2, . . . , n this yields
1≤j 6=l≤n
k(xj , xl) ≥ n ·Dn−1(H),
for each term k(xj , xl) occurs exactly n − 2 times. Now taking the
infimum for all possible x1, . . . , xn ∈ H, we obtain n · Dn(H) ≥ n ·
Dn−1(H), hence the assertion.
The limit D(H) := limn→∞Dn(H) is the transfinite diameter of H.
Similarly, the nth Chebyshev constants converge, too.
PROPOSITION 2. For any H ⊂ X, the Chebyshev constants Mn(H)
converge in the extended sense.
Proof. The sum of two log-polynomials, p(z) =
i=1 k(z, xi) with de-
gree n and q(z) =
j=1 k(z, yj) with degree m, is also a log-polynomial
with degree n+m. Therefore
(n+m)Mn+m ≥ nMn +mMm (3)
for all n,m follows at once. Should Mn(H) be infinity for some n,
then all succeeding terms Mn′(H), n
′ ≥ n are infinity as well, hence
the convergence is obvious. We assume now that Mn(H) is a finite
sequence. At this point, for the sake of completeness, we can repeat the
classical argument of Fekete [8].
Namely, let m,n be fixed integers. Then there exist l = l(n,m) and
r = r(n,m), 0 ≤ r < m nonnegative integers such that n = l ·m + r.
Iterating the previous inequality (3) we get
n ·Mn ≥ l
+ rMr = nMm + r(Mr −Mm).
Fixing now the value of m, the possible values of r remain bounded
by m, and the finitely many values of Mr −Mm’s are finite, too. Hence
dividing both sides by n, and taking lim infn→∞, we are led to
lim inf
Mn ≥ lim inf
Mr −Mm
= Mm .
This holds for any fixed m ∈ N, so taking lim supm→∞ on the right
hand side we obtain
lim inf
Mn ≥ lim sup
that is, the limit exists.
M(H) := limn→∞Mn(H) is called the Chebyshev constant of H.
In the following, we investigate the connection between the Chebyshev
constant M(H) and the transfinite diameter D(H).
THEOREM 3. Let k be a positive, symmetric kernel. For any n ∈ N
and H ⊂ X we have Dn(H) ≤ Mn(H), thus also D(H) ≤ M(H).
Proof. If Mn(H) = +∞, then the assertion is trivial. So assume
Mn(H) < +∞. By the quasi-monotonicity (see (3)) we have that for
all m ≤ n also Mm(H) is finite. We use this fact to recursively find
w1, . . . wn ∈ H such that k(wi, wj) < +∞ for all i < j ≤ n. At the
end we arrive at
1≤i<j≤n k(wi, wj) < +∞, hence Dn(H) < +∞. This
was our first aim to show, in the following this choice of the points
w1, . . . , wn will not play any role. Instead, for an arbitrarily fixed ε > 0,
we take, as we may, an “approximate n-Fekete point system” w1, . . . , wn
(n− 1)n
1≤i 6=j≤n
k(wi, wj) < Dn + ε. (4)
For any x ∈ H the points x,w1, . . . , wn form a point system of n + 1
points, so by the definition of Dn+1 we have
k(x,wi) +
1≤i 6=j≤n
k(wi, wj) ≥ n(n+ 1)Dn+1 ≥ n(n+ 1)Dn,
using also the monotonicity of the sequence Dn. This together with
(4) lead to
pn(x) :=
k(x,wi) ≥
n(n+ 1)
n(n− 1)
Dn + ε
Taking infimum of the left hand side for x ∈ H we obtain
pn(x) ≥ nDn −
n(n− 1)ε
By the very definition of the nth Chebyshev constant, n · Mn ≥
infx∈H pn(x) holds, hence Mn ≥ Dn − (n− 1)ε/2 follows. As this holds
for all ε > 0, we conclude Mn ≥ Dn.
Later we will show that, unlike the classical case of C, the strict
inequality D < M is well possible.
3. Transfinite diameter and energy
We study the connection between the energy w and the transfinite
diameter D. Without assuming the maximum principle we can prove
the equivalence of these two quantities for compact sets. This result
can actually be found in a note of Choquet [4]. There is however a
slight difference to the definitions of Choquet in [4]. There the diagonal
was completely excluded from the definition of D, that is the infimum
in (2) is taken over wi 6= wj, i 6= j and not for systems of arbitrary
wj’s . This means, among others, that in [4] the transfinite diameter is
only defined for infinite sets. The other assumption of Choquet is that
the kernel is infinite on the diagonal. This is completely the contrary
to what we assume in Theorem 8. Indeed, with our definitions of the
transfinite diameter one can even prove equality for arbitrary sets if
the kernel is finite-valued.
THEOREM 4. Let k be an arbitrary kernel and H ⊂ X be any set.
Then D(H) ≤ w(H).
Proof. Let µ ∈ M1(H) be arbitrary, and define ν :=
j=1 µ the
product measure on the product space Xn. We can assume that the
kernel is positive because supp µ, and hence supp ν, is compact so we
can add a constant to k such that it will be positive on these supports.
Consider the following lower semicontinuous functions g and h on Xn
g : (x1, . . . , xn) 7→ Dn(H)
:= inf
(w1,...,wn)∈Xn
n(n−1)
1≤i 6=j≤n
k(wi, wj)
h : (x1, . . . , xn) 7→
n(n−1)
1≤i 6=j≤n
k(xi, xj).
Since 0 ≤ g ≤ h, by the definition of the upper integral the following
holds true
Dn(H) ≤
n(n− 1)
1≤i 6=j≤n
k(xi, xj) dν(x1, . . . , xn)
n(n− 1)
1≤i 6=j≤n
k(xi, xj) dµ(xi) dµ(xj) = W (µ).
Taking infimum in µ yields Dn(H) ≤ w(H), hence also D(H) ≤ w(H).
To establish the converse inequality we need a compactness as-
sumption. With the slightly different terminology, Choquet proves the
following for kernels being +∞ on the diagonal ∆. The arguments there
are very similar, except that the diagonal doesn’t have to be taken care
of in [4]. We give a detailed proof.
PROPOSITION 5 (Choquet [4]). For an arbitrary kernel function k
the inequality D(K) ≥ w(K) holds for all K ⊆ X compact sets.
Proof. First of all the l.s.c. function k attains its infimum on the
compact set K × K. So by shifting k up we can assume that it is
positive, and the validity of the desired inequality is not influenced by
this.
If D(K) = +∞, then by Theorem 4 we have w(K) = +∞, thus
the assertion follows. Assume therefore D(K) < +∞, and let n ∈ N,
ε > 0 be fixed. Let us choose a Fekete point system w1, . . . , wn from
K. Put µ := µn := 1/n
i=1 δwi where δwi are the Dirac measures at
the points wi, i = 1, . . . , n. For a continuous function 0 ≤ h ≤ k with
compact support, we have
h dµ dµ =
i,j=1
h(wi, wj)
h(wi, wi) +
i,j=1
h(wi, wj)
h(wi, wi) +
i,j=1
k(wi, wj)
i,j=1
k(wi, wj)
Dn(K) ≤
+D(K)
using, in the last step, also the monotonicity of the sequenceDn (Propo-
sition 1). In fact, we obtain for n ≥ N = N(‖h‖, ε) the inequality
h dµ dµ ≤ D + ε. (5)
It is known, essentially by the Banach-Alaoglu Theorem, that for a
compact set K the measures of M1(K) form a weak
∗-compact subset
of M, hence there is a cluster point ν ∈ M1(K) of the set MN :=
{µn : n ≥ N} ⊂ M1(K). Let {να}α∈I ⊆ MN be a net converging to
ν. Recall that να⊗να weak
∗-converges to ν⊗ν. We give the proof. For
a function g ∈ C(K ×K), g(x, y) = g1(x) · g2(y) it is obvious that
g dνα dνα →
g dν dν. (6)
The set A of such product-decomposable functions g(x, y) = g1(x)g2(y)
is a subalgebra of C(K ×K), which also separates X ×X, since it is
already coordinatewise separating. By the Stone–Weierstraß theorem
A is dense in C(K ×K). From this, using also that the family MN of
measures is norm-bounded, we immediately get the weak∗-convergence
(6). All these imply
h dν dν ≤ D(K) + ε,
w(K) ≤ W (ν) :=
kdνdν = sup
0 ≤ h ≤ k
h ∈ Cc(X ×X)
hdνdν ≤ D(K)+ε,
for all ε > 0. This shows w(K) ≤ D(K).
COROLLARY 6 (Choquet [4]). For arbitrary kernel k and compact set
K ⊂ X, the equality D(K) = w(K) holds.
Proof. By compactness we can shift k up and therefore assume it is
positive. Then we apply Theorem 4 and Proposition 5.
The assumptions of Choquet [4] are the compactness of the set plus the
property that the kernel is +∞ on the diagonal (besides it is continuous
in the extended sense). This ensures, loosely speaking, that for a set K
of finite energy an energy minimising measure µ (i.e., for whichW (µ) =
w(K)) is necessarily non-atomic, moreover µ ⊗ µ is not concentrated
on the diagonal. Therefore to show equality of w with D, one has to
exclude the diagonal completely from the definition of the transfinite
diameter.
We however allow a larger set of choices for the point system in the
definition of D. Indeed, we allow Fekete points to coincide, and this also
makes it possible to define the transfinite diameter of finite sets. With
this setup the inequality D ≤ w is only simpler than in the case handled
by Choquet. Whereas, however surprisingly, the equality D(K) = w(K)
is still true for compact sets K but without the assumption on the
diagonal values of the kernel.
We will see in §5 Example 13 that even assuming the maximum prin-
ciple but lacking the compactness allows the strict inequality D < w.
This phenomena however may exist only in case of unbounded kernels,
as we will see below. In fact, we show that if the kernel is finite on the
diagonal, thenD = w holds for arbitrary sets. For this purpose, we need
the following technical lemma, which shows certain inner regularity
properties of D and is also interesting in itself.
LEMMA 7. Assume that the kernel k is positive and finite on the
diagonal, i.e., k(x, x) < +∞ for all x ∈ X. Then for an arbitrary
H ⊂ X we have
D(H) = inf
K ⊂ H
K compact
D(K) = inf
W ⊂ H
#W < ∞
D(W ). (7)
Proof. The inequality infD(K) ≤ infD(W ) is clear. For H ⊇ K the
inequality D(H) ≤ D(K) is obvious, so we can assume D(H) < +∞.
For ε > 0 let W = {w1, . . . , wn} be an approximate n-Fekete point set
of H satisfying (4). Then
D(W ) = lim
Dmn(W ) ≤ lim
mn(mn− 1)
1≤i′ 6=j′≤mn
k(wi′ , wj′),
where
wi′ :=
. . .
′ = i+ rn, r = 0, . . . ,m− 1
. . .
Set C := max{k(x, x) : x ∈ W}. So we find
D(W ) ≤ lim
mn(mn−1)
1≤i 6=j≤n
k(wi, wj) +
mn(mn−1)
1≤i≤n
k(wi, wi)
1≤i 6=j≤n
k(wi, wj) lim
mn(mn−1)
+ Cn lim
mn(mn−1)
1≤i 6=j≤n
k(wi, wj) ≤
(Dn(H) + ε) ≤ D(H) + ε.
This being true for all ε > 0, taking infimum we finally obtain
W ⊂ H
#W < ∞
D(W ) ≤ D(H).
Clearly, if k(x, x) = +∞ for all x ∈ W with a finite set #W = n,
then for all m > n we have Dm(W ) = +∞. Thus in particular for
kernels with k : ∆ → {+∞}, the above can not hold in general, at least
as regards the last part with finite subsets.
Now, completely contrary to Choquet [4] we assume that the kernel
is finite on the diagonal and prove D = w for any set. Hence an example
of D < w (see §5 Example 13) must assume k(x, x) = +∞ at least for
some point x.
THEOREM 8. Assume that the kernel k is positive and is finite on
the diagonal, that is k(x, x) < +∞ for all x ∈ X. Then for arbitrary
sets H ⊂ X, the equality D(H) = w(H) holds.
Proof. By Theorem 4 we have D(H) ≤ w(H). Hence there is nothing
to prove, if D(H) = +∞. Assume D(H) < +∞, and let ε > 0 be
arbitrary. By Lemma 7 we have for some n ∈ N a finite set W =
{w1, w2 . . . , wn} with D(H) + ε ≥ D(W ). In view of Proposition 5
we have D(W ) ≥ w(W ), and by monotonicity also w(W ) ≥ w(H). It
follows that D(H) + ε ≥ w(H) for all ε > 0, hence also the “≥” part
of the assertion follows.
4. Energy and Chebyshev constant
To investigate the relationship between the energy and the Cheby-
shev constant the following general version of Frostman’s Equilibrium
Theorem [9, Theorem 2.4] is fundamental for us.
THEOREM 9 (Fuglede). Let k be a positive, symmetric kernel and
K ⊂ X be a compact set such that w(K) < +∞. Every µ which
has minimal energy (µ ∈ M1(K),W (µ) = w(K)) satisfy the following
properties
Uµ(x) ≥ w(K) for nearly every1 x ∈ K,
Uµ(x) ≤ w(K) for every x ∈ supp µ,
Uµ(x) = w(K) for µ-almost every x ∈ X.
Moreover, if the kernel is continuous, then
Uµ(x) ≥ w(K) for every x ∈ K.
THEOREM 10. Let H ⊂ X be arbitrary. Assume that the kernel k is
positive, symmetric and satisfies the maximum principle. Then we have
Mn(H) ≤ w(H) for all n ∈ N, whence also M(H) ≤ w(H) holds true.
Proof. Let n ∈ N be arbitrary. First let K be any compact set.
We can assume w(K) < +∞, since otherwise the inequality holds
irrespective of the value of Mn(K). Consider now an energy-minimising
measure νK of K, whose existence is assured by the lower semicontinu-
ity of µ 7→
k dµ dµ and the compactness of M1(K), see [9, Theorem
2.3].
By the Frostman-Fuglede theorem (Theorem 9) we have UνK (x) ≤
w(K) for all x ∈ supp νK , so V (νK) ≤ w(K), and by the maximum
principle even
UνK (x) ≤ w(K) for all x ∈ X.
1 The set A of exceptional points is small in the sense w(A) = +∞.
Then for all w1, . . . , wn ∈ K
k(x,wj) ≤
k(x,wj) dνK(x) ≤ w(K) .
Taking supremum for w1, . . . , wn ∈ K, we obtain
w1,...,wn∈K
k(x,wj) ≤ w(K).
So Mn(K) ≤ w(K) for all n ∈ N.
Next let H ⊂ X be arbitrary. In view of the last form of (1), for all
ε > 0 there exists a measure µ ∈ M1(H), compactly supported in H,
with w(µ) ≤ w(H) + ε. Let W = {w1, . . . , wn} ⊂ H be arbitrary and
define pW (x) :=
i k(x,wi).
Consider the compact set K := W ∪ supp µ ⊂ H. By definition of
the energy, supp µ ⊂ K implies w(K) ≤ w(µ), hence w(K) ≤ w(H) +
ε. Combining this with the above, we come to Mn(K) ≤ w(H) + ε.
Since W ⊂ K, by definition of Mn(K) we also have
pW (x) ≤ Mn(K). (8)
The left hand side does not increase, if we extend the inf over the
whole of H, and the right hand side is already estimated from above
by w(H) + ε. Thus (8) leads to
pW (x) ≤ w(H) + ε.
This holds for all possible choices of W = {w1, . . . , wn} ⊂ H, hence is
true also for the sup of the left hand side. By definition of Mn(H) this
gives exactly Mn(H) ≤ w(H) + ε, which shows even Mn(H) ≤ w(H).
Remark. In [6] it is proved that M(H) = q(H), where
q(H) = inf
µ∈M1(H)
Uµ(x).
The idea behind is a minimax theorem, see also [16, 17]. Trivially
w(H) ≤ q(H) ≤ u(H). So the maximum principle implies M(H) =
w(H) = q(H) = u(H).
5. Summary of the Results. Examples
In this section, we put together the previous results, thus proving the
equality of the three quantities being studied, under the assumption
of the maximum principle for the kernel. Further, via several instruc-
tive examples we investigate the necessity of our assumptions and the
sharpness of the results.
THEOREM 11. Assume that the kernel k is positive, symmetric and
satisfies the maximum principle. Let K ⊂ X be any compact set. Then
the transfinite diameter, the Chebyshev constant and the energy of K
coincide:
D(K) = M(K) = w(K).
Proof. We presented a cyclic proof above, consisting of M ≥ D
(Theorem 3), D ≥ w (Proposition 5) and finally w ≥ M (Theorem 10).
THEOREM 12. Assume that the kernel k is positive, finite and sat-
isfies the maximum principle. For an arbitrary subset H ⊂ X the
transfinite diameter, the Chebyshev constant and the energy of H co-
incide:
D(H) = M(H) = w(H).
Proof. By finiteness D = w, due to Theorem 8. This with D ≤ M
and M ≤ w (Theorems 3 and 10) proves the assertion.
Remark. In the above theorem, logically it would suffice to assume
that the kernel be finite only on the diagonal. But if this was the case,
the maximum principle would then immediately imply the finiteness of
the kernel everywhere.
Let us now discuss how sharp the results of the preceding sections
are. In the first example we show that, if we drop the assumption of
compactness the assertions of Theorem 3, Theorem 4 and Theorem 10
are in general the strongest possible.
Example 13. Let X = N ∪ {0} endowed with discrete topology and
the kernel
k(n,m) :=
+∞ if n = m,
0 if 0 6= n 6= m 6= 0,
1 otherwise.
The kernel is symmetric, l.s.c. and has the maximum principle. This
latter can be seen by noticing that for a probability measure µ ∈ M1(X)
the potential is +∞ on the support of µ. Indeed, since X is countable,
all measures µ ∈ M1(X) are necessarily atomic, and if for some point
ℓ ∈ X we have µ({ℓ}) > 0, then by definition
X k(x, y) dµ(y) = +∞.
We calculate the studied quantities of the set H = X (also as in all
the examples below). Since the kernel is positive, Dn ≥ 0. On the other
hand, choosing w1 := 1, . . . , wn := n, all the values k(wi, wj) will be
exactly 0, so it follows that Dn = 0, n = 1, 2, . . ., and hence D = 0.
The Chebyshev constant can be estimated from below, if we compute
the infimum of a suitably chosen log-polynomial. Consider the log-
polynomial p(x) with all zeros placed at 0, that is with w1 = . . . =
wn = 0. Then the log-polynomial p(x) is
j k(x,wj) = n · k(x, 0). If
x 6= 0, we have p(x) = n, which gives M ≥ 1. The upper estimate of
M is also easy: suppose that in the system w1, . . . wn there are exactly
m points being equal to 0 (say the first m). Then
p(x) =
+∞ x = w1, . . . , wn,
n x = 0, x 6= w1, . . . , wn (if m = 0)
m x 6= 0, x 6= w1, . . . , wn
This shows for the corresponding log-polynomial inf p(x) = m, so
Mn ≤ 1, whence M = 1.
The energy is computed easily. Using the above reasoning on the
maximum principle, we see W (µ) = +∞ for any µ ∈ M1(X), hence
w(X) = +∞.
Thus we have an example of
+∞ = w > M > D = 0.
The above example completes the case of the kernel with maximum
principle. Let us now drop this assumption and look at what can
happen.
Example 14. Let X := {−1, 0, 1} be endowed with the discrete topol-
ogy. We define the kernel by
k(x, y) :=
2 if 0 ≤ |x− y| < 2,
0 if 2 = |x− y|.
Then k is continuous and bounded on X×X. This, in any case, implies
D = w by Theorem 8. Note that k does not satisfy the maximum
principle. To see this, consider, e.g., the measure µ = 1
δ1. Then
for the potential Uµ one has Uµ(1) = Uµ(−1) = 1 and Uµ(0) = 2,
which shows the failure of the maximum principle.
To estimate the nth diameter from above, let us consider the point
system {wi} of n = 2m points with m points falling at −1 and m points
falling at 1, while no points being placed at 0. Then by definition of
Dn := Dn(X) one can write
n(n− 1)
Dn ≤ 2
· 2 +m2 · 0 =
Applying this estimate for all even n = 2m as n → ∞, it follows that
D = lim
Dn ≤ 1. (9)
Next we estimate the Chebyshev constants from below by computing
the infimum of some special log-polynomials. For pn(x) = k(x, 0) one
has pn(x) ≡ 2 = inf pn. We thus find Mn ≥ 2 and M ≥ 2, showing
M > D, as desired.
Example 15. Let X := N with the discrete topology. Then X is
a locally compact Hausdorff space, and all functions are continuous,
hence l.s.c. on X. Let k : X ×X → [0,+∞] be defined as
k(n,m) :=
+∞ if n = m,
2−n−m if n 6= m.
Clearly k is an admissible kernel function. For the energy we have
again w(X) = +∞, see Example 13.
On the other hand let n ∈ N be any fixed number, and compute the
nth diameter Dn(X). Clearly if we choose wj := m+ j, with m a given
(large) number to be chosen, then we get
Dn(H) ≤
(n− 1)n
1≤i 6=j≤n
2−i−j−2m ≤
(n− 1)n
≤ 2−2m ,
hence we find that the nth diameter is Dn(X) = 0, so D(X) = 0,
too. For any log-polynomial p(x) we have inf p(x) = limx→∞ p(x) = 0,
hence M(X) = 0. That is we have D(X) = M(X) = 0 < w(X) = +∞.
The example shows how important the diagonal, excluded in the
definition of D but taken into account in w, may become for particular
cases. We can even modify the above example to get finite energy.
Example 16. Let X := (0, 1], equipped with the usual topology, and
let xn = 1/n. We take now
k(x, y) :=
+∞ if x = y,
2−n−m if x = xn and y = xm (xn 6= xm),
− log |x− y| otherwise
Compared to the l.s.c. logarithmic kernel, this k assumes different,
smaller values at the relatively closed set of points {(xn, xm) : n 6=
m} ⊂ X ×X only, hence it is also l.s.c. and thus admissible as kernel.
If a measure µ ∈ M1(X) has any atom, say if for some point z ∈ X
we have µ({z}) > 0, then by definition
X k(x, y) dµ(y) = +∞, hence
also w(µ) = +∞. Since for all µ ∈ M1(X) with any atomic component
w(µ) = +∞, we find that for the set H := X we have
w(H) := inf
µ∈M1(H)
w(µ) = inf
µ∈M1(H)
µ not atomic
w(µ).
But for measures without atoms, the countable set of the points xn are
just of measure zero, hence the energy equals to the energy with respect
to the logarithmic kernel. Thus we conclude w(H) = e−cap(H) = e−1/4,
as cap((0, 1]) = 1/4 is well-known.
On the other hand if n ∈ N is any fixed number, we can compute
the nth diameter Dn(H) exactly as above in Example 15. Hence it is
easy to see that Dn(H) = 0, whence also D(H) = 0. Similarly, we find
M(H) = 0, too.
This example shows that even in case w(H) < +∞ we can have
w(H) > D(H) = M(H).
6. Average distance number and the maximum principle
In the previous section, we showed the equality of the Chebyshev con-
stant M and the transfinite diameter D, using essentially elementary
inequalities and the only theoretically deeper ingredient, the assump-
tion of the maximum principle. We have also seen examples showing
that the lack of the maximum principle for the kernel allows strict
inequality between M and D. These observations certify to the rel-
evance of this principle in our investigations. Indeed, in this section
we show the necessity of the maximum principle in case of continuous
kernels for having M(K) = D(K) for all compact sets K. We need
some preparation first.
Recall from the introduction the notion of the average distance (or
rendezvous) number. Actuyally, a more general assertion than there can
be stated, see Stadje [21] or [6]. For a compact connected set K and a
continuous, symmetric kernel k, the average distance number r(K) is
the uniquely existing number with the property that for all probability
measures supported in K there is a point x ∈ K with
Uµ(x) =
k(x, y) dµ(y) = r(K).
This can be even further generalised by dropping the connectedness,
see Thomassen [22] and [6]. Even for not necessarily connected but
compact spacesK with symmetric, continuous kernel k there is a unique
number r(K) with the property that whenever a probability measure
on K and a positive ε are given, there are points x1, x2 ∈ K such that
Uµ(x1)− ε ≤ r(K) ≤ U
µ(x2) + ε.
This number is called the (weak) average distance number, and is par-
ticularly easy to calculate, when a probability measure with constant
potential is available. Such a measure µ is called then an invariant
measure. In this case the average distance number r(K) is trivially just
the constant value of the potential Uµ, see Morris, Nicholas [14] or [7].
It was proved in [7] that one always has M(K) = r(K), so once we
have an invariant measure, then the Chebyshev constant is again easy
to determine.
Also the Wiener energy w(K) has connection to invariant measures,
as shown by the following result, which is a simplified version of a more
general statement from [7], see also Wolf [23].
THEOREM 17. Let ∅ 6= K ⊂ X be a compact set and k be a continu-
ous, symmetric kernel. Then we have
r(K) ≥ w(K).
Furthermore, if r(K) = w(K), then there exists an invariant measure
in M1(K).
As mentioned above, we have r(K) = M(K), so the inequality r(K) ≥
w(K) in the first assertion of the above theorem is also the conse-
quence of Theorems 3 and 8. For the proof of the second assertion
one can use the Frostman-Fuglede Equilibrium Theorem 9 with the
obvious observation that “nearly every” in this context means indeed
“every”. Actually any probability measure µ ∈ M1(K) which minimises
ν 7→ supK U
ν is an invariant measure and its potential is constant
M(K), see [7, Thm. 5.2] (such measures undoubtedly exist because of
compactness of M1(K)). Henceforth we will indifferently use the terms
energy minimising or invariant for expressing this property of measures.
THEOREM 18. Suppose that the kernel k is symmetric and continu-
ous. If M(K) = D(K) for all compact sets K ⊆ X, then the kernel has
the maximum principle.
Proof. Recall from Corollary 6 that D(K) = w(K) for all K ⊆ X
compact. So we can use Theorem 17 all over in the following arguments.
We first prove the assertion in the case when X is a finite set. The
proof is by induction on n = #X. For n = 1 the assertion is trivial.
Let now #X = 2, X = {a, b}. Assume without loss of generality that
k(a, a) ≤ k(b, b). Then we only have to prove that for µ = δa the
maximum principle, i.e., the inequality k(a, b) ≤ k(a, a) holds. To see
this we calculate M(X) and D(X). We certainly have D(X) ≤ k(a, a).
On the other hand for an energy minimising probability measure νp :=
pδa + (1 − p)δb on X we know that its potential is constant over X,
hence
pk(a, a) + (1− p)k(b, a) = pk(a, b) + (1− p)k(b, b)
= M(X) = D(X) ≤ k(a, a).
Here if p = 1, then k(a, a) = k(a, b). If p < 1, then we can write
(1− p)k(b, a) ≤ (1− p)k(a, a), hence k(b, a) ≤ k(a, a),
so the maximum principle holds.
Assume now that the assertion is true for all sets with at most n
elements and for all kernels, and let #X = n + 1. For a probability
measure µ on X we have to prove supx∈X U
µ(x) = supx∈ supp µ U
µ(x).
If supp µ = X, then there is nothing to prove. Similarly, if there are
two distinct points x1 6= x2, x1, x2 ∈ X \ supp µ, then by the induction
hypothesis we have
x∈X\{x1}
Uµ(x) = sup
x∈ supp µ
Uµ(x) = sup
x∈X\{x2}
Uµ(x).
So for a probability measure µ defying the maximum principle we
must have # supp µ = n, say supp µ = X \ {xn+1}; let µ be such a
measure. Set K = supp µ and let µ′ be an invariant measure on K.
We claim that all such measures µ′ are also violating the maximum
principle. If µ = µ′, we are done. Assume µ 6= µ′ and consider the
linear combinations µt := tµ+(1− t)µ
′. There is a τ > 1, for which µτ
is still a probability measure and supp µτ ( supp µ. By the inductive
hypothesis (as # supp µτ < n) we have U
µτ (xn+1) ≤ U
µτ (a) for some
a ∈ supp µτ . We also know that U
µ(xn+1) = U
µ1(xn+1) > U
µ1(a).
Hence for the linear function Φ(t) := Uµt(xn+1) − U
µt(a) we have
Φ(1) > 0 and also Φ(τ) ≤ 0 (τ > 1). This yields Φ(0) > 0, i.e.,
(xn+1) = U
µ0(xn+1) > U
µ0(a) = Uµ
(y) for all y ∈ K. We have
therefore shown that all energy minimising (invariant) measures on K
must defy the maximum principle.
Let now ν be an invariant measure on X. We have
M(X) = Uν(y) = sup
Uν(x) = D(X)
≤ D(K) = sup
(x) = Uµ
(z) < Uµ
(xn+1)
for all y ∈ X, z ∈ K. Thus we can conclude Uν(y) ≤ Uµ
(y) for all
y ∈ X and even “<” for y = xn+1. Integrating with respect to ν would
yield
k dν dν = M(X) <
k dµ′ dν =
k dν dµ′ = M(X),
hence a contradiction, unless ν({xn+1}) = 0. If ν({xn+1}) = 0 held,
then ν would be an energy minimising measure on K. This is because
obviously supp ν ⊆ K holds, and the potential of ν is constant M(X)
over K, so
M(X) =
k dν dµ′ =
k dµ′ dν = M(K) holds.
As we saw above, then ν would not satisfy the maximum principle,
a contradiction again, since the potential of ν is constant on X. The
proof of the case of finite X is complete.
We turn now to the general case of X being a locally compact space
with continuous kernel. Let µ be a compactly supported probability
measure on X and y 6∈ supp µ. Set K = supp µ and note that both
M1(K) ∋ ν 7→ supK U
ν and ν 7→ Uν(y) are continuous mappings with
respect to the weak∗-topology on M1(K). If supK U
µ < Uµ(y) were
true, we could therefore find, by a standard approximation argument,
see for example [6, Lemma 3.8], a finitely supported probability measure
µ′ on K for which
x∈ supp µ′
(x) ≤ sup
(x) < Uµ
This is nevertheless impossible by the first part of the proof, thus the
assertion of the theorem follows.
Acknowledgement
The authors are deeply indebted to Szilárd Révész for his insightful
suggestions and for the motivating discussions.
References
1. Anagnostopoulos, V. and Sz. Gy. Révész: 2006, ‘Polarization constants for
products of linear functionals over R2 and C2 and Chebyshev constants of
the unit sphere’. Publ. Math. Debrecen 68(1–2), 75–83.
2. Bourbaki, N.: 1965, Intégration, Éléments de Mathématique XIII., Vol. 1175 of
Actualités Sci. Ind. Paris: Hermann, 2nd edition.
3. Carleson, L.: 1967, Selected Problems on Exceptional Sets, Vol. 13 of Van
Nostrand Mathematical Studies. D. Van Nostrand Co., Inc.
4. Choquet, G.: 1958/59, ‘Diamètre transfini et comparaison de diverses ca-
pacités’. Technical report, Faculté des Sciences de Paris.
5. Farkas, B. and Sz. Gy. Révész: 2005, ‘Rendezvous numbers in normed spaces’.
Bull. Austr. Math. Soc. 72, 423–440.
6. Farkas, B. and Sz. Gy. Révész: 2006a, ‘Potential theoretic approach to
rendezvous numbers’. Monatshefte Math 148, 309–331.
7. Farkas, B. and Sz. Gy. Révész: 2006b, ‘Rendezvous numbers of metric spaces
– a potential theoretic approach’. Arch. Math. (Basel) 86, 268–281.
8. Fekete, M.: 1923, ‘Über die Verteilung der Wurzeln bei gewissen algebraischen
Gleichungen mit ganzahligen Koeffizienten’. Math. Z. 17, 228–249.
9. Fuglede, B.: 1960, ‘On the theory of potentials in locally compact spaces’. Acta
Math. 103, 139–215.
10. Fuglede, B.: 1965, ‘Le théorème du minimax et la théorie fine du potentiel’.
Ann Inst. Fourier 15, 65–87.
11. Gross, O.: 1964, ‘The rendezvous value of a metric space’. Ann. of Math. Stud.
52, 49–53.
12. Landkof, N. S.: 1972, Foundations of modern potential theory, Vol. 180 of
Die Grundlehren der mathematischen Wissenschaften. New York, Heidelberg:
Springer.
13. Mhaskar, H. N. and E. B. Saff: 1992, ‘Weighted analogues of capacity,
transfinite diameter and Chebyshev constants’. Constr. Approx. 8(1), 105–124.
14. Morris, S. A. and P. Nickolas: 1983, ‘On the average distance property of
compact connected metric spaces’. Arch. Math. 40, 459–463.
15. Ohtsuka, M.: 1961, ‘On potentials in locally compact spaces’. J. Sci. Hiroshima
Univ. ser A 1, 135–352.
16. Ohtsuka, M.: 1965, ‘An application of the minimax theorem to the theory of
capacity’. J. Sci. Hiroshima Univ. ser A 29, 217–221.
17. Ohtsuka, M.: 1967, ‘On various definitions of capacity and related notions’.
Nagoya Math. J. 30, 121–127.
18. Pólya, Gy. and G. Szegő: 1931, ‘Über den transfiniten Durchmesser (Ka-
pazitätskonstante) von ebenen und räumlichen Punktmengen’. J. Reine Angew.
Math. 165, 4–49.
19. Révész, Sz. Gy. and Y. Sarantopoulos: 2004, ‘Plank problems, polarization,
and Chebyshev constants’. J. Korean Math. Soc. 41(1), 157–174.
20. Saff, E. B. and V. Totik: 1997, Logarithmic potentials with external fields, Vol.
316 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin.
21. Stadje, W.: 1981, ‘A property of compact, connected spaces’. Arch. Math. 36,
275–280.
22. Thomassen, C.: 2000, ‘The rendezvous number of a symmetric matrix and a
compact connected metric space’. Amer. Math. Monthly 107(2), 163–166.
23. Wolf, R.: 1997, ‘On the average distance property and certain energy integrals’.
Ark. Mat. 35, 387–400.
24. Zaharjuta, V. P.: 1975, ‘Transfinite diameter, Chebishev constants, and
capacity for compacta in Cn’. Math. USSR Sbornik 25(3), 350–364.
