In this article we study the analog of Kakeya needle problem for $(n-1)$-rectifiable sets in $\mathbb{R}^n$ and construct the related Nikodym type sets. The novelty of our approach lies in combining three ingredients: the two-dimensional Venetian blind-type construction for isometries in $\mathbb{R}^n$; the normal geometry of a $(n-1)$-rectifiable set viewed in the projective setting; and measure estimates of moving $(n-1)$-rectifiable sets by isometries in $\mathbb{R}^n$. Together, these ingredients enable us to move $(n-1)$-rectifiable sets along paths of isometries in $\mathbb{R}^n$ and cover a Lebesgue null set.
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)\ge \frac{\operatorname{per}K}{6}$ for every planar convex body.
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $\epsilon$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This cover is of size $n^{O(\log(mn)/\epsilon^2)}$, provided the polynomials have constant range over the smallest $\ell_1$-ball inscribing ${H}$. Our approach extends classical net-based sparsifications for linear functions (e.g., Lipton, Markakis, and Mehta [2003]) to arbitrary families of constant-degree polynomials over general convex sets. We use a two-step scheme: first, we construct a quasi-polynomial pre-cover of the family on the smallest $\ell_1$-ball containing ${H}$ by using a concentration argument and leveraging a connection between Bernstein approximation and multinomial distributions; we then compress the pre-cover to ${H}$ by using a recursive degree reduction and feasibility programs anchored at points of the pre-cover. The existence of these covers immediately yields a unified framework for Quasi-Polynomial Time Approximation Schemes (QPTAS) across a wide range of a problems, including fixed-degree polynomial minimization over polyhedral sets, Constraint Satisfaction Problems (CSPs), Free Games, variational inequalities with polynomial operators (which implies guarantees for local Nash equilibria in polynomial games), and additive approximation for normalized densest $k$-subhypergraph on $O(1)$-uniform hypergraphs.
Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.· 0 citations
We continue the study of the natural polytope $\mathcal{D}$ in $\mathbb{R}^{n+d}$ associated with the disjunction of a set of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary variables. Already $\mathcal{D}$ had been characterized for arbitrary $n\geq 1$ and (i) $d\in\{1,2\}$, and (ii) for a broad generalization of hyper-rectangles. In both cases, the complete characterization employs full optimal big-M lifting. Here, we give a complete description of $\mathcal{D}$ for the case of $n=1$ and arbitrary $d$, when the (two) polytopes are arbitrary generalized cross polytopes. Furthermore, we characterize when our complete description employs only optimal big-M lifting. For $n>1$, we generalize the family of facet-describing inequalities used for $n=1$. Finally, we carry out some computational experiments demonstrating the value of our theoretical results.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erd\H{o}s conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by S\'ark\"ozy and later again by Ruzsa, who found an elegant construction showing that $D(N)\geq c\cdot N^{0.733077\dots}$, with an absolute constant $c>0$. His approach was subsequently refined, leading to the previously best known lower bound with exponent $0.7334117\dots$ due to Beigel-Gasarch and, independently, Lewko. However, in the original paper Ruzsa observed that $3/4$ seems to be the natural barrier of his approach. In this paper we develop a new construction leading to the lower bound \[ \liminf_{N\to\infty}\frac{\log D(N)}{\log N} \geq \alpha_*:= 0.7527964558\ldots; \] thus crossing the natural exponent-$3/4$ barrier of Ruzsa's method. The value $0.7527964558\ldots$ arises from a simple optimisation problem and appears to be the limit of the new approach.