ﻻ يوجد ملخص باللغة العربية
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $alpha$ and a convex body $B$, $g_{alpha}(B)$ is the infimum of $alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $alpha$. In this paper, we prove that $g_{alpha}(B)leq h_{alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{alpha} (B) > 2^{d-alpha}$ for almost all $alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
Given a Borel measure $mu$ on ${mathbb R}^{n}$, we define a convex set by [ M({mu})=bigcup_{substack{0le fle1, int_{{mathbb R}^{n}}f,{rm d}{mu}=1 } }left{ int_{{mathbb R}^{n}}yfleft(yright),{rm d}{mu}left(yright)right} , ] where the union is taken ov
Let $K$ be a convex body in $mathbb{R}^n$ and $f : partial K rightarrow mathbb{R}_+$ a continuous, strictly positive function with $intlimits_{partial K} f(x) d mu_{partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmet
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the othe
We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bo
Barker and Larman asked the following. Let $K subset {Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K subset {Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K cap