We consider the trajectories of points on under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is d+1.
A standard technique for establishing geometric inequalities is to approximate full radial symmetrization by a sequence of simpler symmetrizations. Two-point symmetrization has been used in this way to prove the isoperimetric inequality on spheres @cite_12 , and sharp inequalities for path integrals @cite_8 @cite_3 @cite_2 . The convergence of random sequences of symmetrizations has received some attention in the literature, most notably in the work of Klartag @cite_4 on Steiner symmetrizations of convex sets. Convergence of two-point symmetrizations is less well studied. Very recently, De Keyser and Van Schaftingen have considered random symmetrization processes with time correlations @cite_14 . Open questions in the area include precise conditions for convergence, and bounds on the rate of convergence.
@cite_14 @cite_4 @cite_8 @cite_3 @cite_2 @cite_12
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Under continuity and recurrence assumptions, we prove that the iteration of successive partial symmetrizations that form a time-homogeneous Markov process, converges to a symmetrization. We cover several settings, including the approximation of the spherical nonincreasing rearrangement by Steiner symmetrizations, polarizations and cap symmetrizations. A key tool in our analysis is a quantitative measure of the asymmetry. It is a classical fact, that given an arbitrary convex body (K R ^n , ) there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also make use of an earlier result by the author regarding “isomorphic Minkowski symmetrization”. We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of constant curvature in terms of the exit time from a disc of the same volume. This amounts to a rearrangement inequality for the Dirichlet heat kernel. To connect this inequality with the classical isoperimetric inequality, we derive a formula for the perimeter of a set in terms of the heat flow over the boundary. An auxiliary result generalizes Riesz' rearrangement inequality to multiple integrals. Let (ξ(s)) s ≥ 0 be a standard Brownian motion in d ≥ 1 dimensions and let (D s ) s ≥ 0 be a collection of open sets in ( R ^d ). For each s, let B s be a ball centered at 0 with vol(B s ) = vol(D s ). We show that ( E [ vol ( s t ( (s) + D_s))] E [ vol ( s t ( (s) + B_s))] ), for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion. where de denotes normalized surface measure, V is the conformal gradient and q = (2n) (n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see [8]). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is "What is the situation in dimension 2?" Two important arguments ([25], [26], [27]) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in part