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 conje...
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...
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...
Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.· arXiv.org· 1 citation
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 generalizatio...
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...
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...
D. Krachun· 1 citation
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