We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
Shortly before uploading this paper we came across a very recent paper @cite_5 on arXiv which investigates related questions and proves a variant of Theorem for sets @math of sufficiently small measure (Proposition 11, @cite_5 ), as well as much more. Their proof uses a covering lemma based on a result of @cite_2 , which is no longer true in higher dimensions. Our proof is different and we believe can be adapted to work in higher dimensions as well. Moreover, the stretching result only for sufficiently small sets is not sufficient for our purposes in proving rough isometry. Our work was independent of @cite_5 .
@cite_5 @cite_2
2963702702 2001663593 2609098951 2963780857
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes).