Related papers: New estimates for $d_{2,1}$ and $d_{3,2}$

Let $K$ be a centrally-symmetric convex body in $\mathbb{R}^n$ and let $\|\cdot\|$ be its induced norm on ${\mathbb R}^n$. We show that if $K \supseteq r B_2^n$ then: \[ \sqrt{n} M(K) \leqslant C \sum_{k=1}^{n} \frac{1}{\sqrt{k}}…

Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density…

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that…

It is known that for a convex body K in R^d of volume one, the expected volume of random simplices in K is minimised if K is an ellipsoid, and for d = 2, maximised if K is a triangle. Here we provide corresponding stability estimates.

Let $\gamma^d_m(K)$ be the smallest positive number $\lambda$ such that the convex body $K$ can be covered by $m$ translates of $\lambda K$. Let $K^d$ be the $d$-dimensional crosspolytope. It will be proved that $\gamma^d_m(K^d)=1$ for…

Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex…

For every convex body $K \subset \mathbb R^n$ and $\delta \in (0,1)$, the $\delta$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq \delta \left|K\right|_n$. We show that for $n=2$ and…

For integers $k,t \geq 2$ and $1\leq r \leq t$ let $D_k(r,t;n)$ be the number of parts among all $k$-regular partitions (i.e., partitions of $n$ where all parts have multiplicity less than $k$) of $n$ that are congruent to $r$ modulo $t$.…

The distance between convex bodies \(K, L \subseteq \R^n\) is defined as \[ d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\}, \] where the infimum is taken over all \(x,y \in \R^n\) and all…

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies…

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the…

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <=…

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation…

Let $\varphi_{n,K}$ denote the largest angle in all the triangles with vertices among the $n$ points selected at random in a compact convex subset $K$ of $\mathbb{R}^d$ with nonempty interior, where $d\ge2$. It is shown that the…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $\Lambda$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional…

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…

A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and $\gamma_{n}(K) \leq 1/2$ there is $\gamma_{n}(sK) \leq (2s)^{w^{2}/4}\gamma_{n}(K)$ for any…

Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed as special cases of the classes of k-monotone densities. A density g is said to…