Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

Illuminating and covering convex bodies

Covering numbers of convex bodies based on homothetical copies and related illumination numbers are well-known in combinatorial geometry and, for example, related to Hadwiger's famous covering problem. Similar numbers can be defined by using proper translates instead of homothets, and even more related concepts make sense. On these lines we introduce some new covering and illumination numbers of convex bodies, present their properties and compare them with each other as well as with already known numbers. Finally, some suggestive examples illustrate that these new illumination numbers are inte…

Illuminating spiky balls and cap bodies

The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of $2$-illuminable spiky balls as well as centrally symmetric cap bodies. In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions by showing that any $d$-dimensional centrally symmetric cap body can be illuminated by $<2^d$ directions in Euclidean $d$-space for all $d\geq 20$. Further…

关于照亮常宽体的注记

Note on illuminating constant width bodies

本研究推广已有构造方法，改进了$\mathbb{R}^n$中常宽体照明数的指数下界，得到至少为$(τ+o(1))^n$（$τ\approx1.047$）的新结果。

Volumetric bounds for intersections of congruent balls

We investigate the intersections of balls of radius $r$, called $r$-ball bodies, in Euclidean $d$-space. An $r$-lense (resp., $r$-spindle) is the intersection of two balls of radius $r$ (resp., balls of radius $r$ containing a given pair of points). We prove that among $r$-ball bodies of given volume, the $r$-lense (resp., $r$-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of $r$-ball bodies of given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for $r$…

Covering by homothets and illuminating convex bodies

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $α$ and a convex body $B$, $g_α(B)$ is the infimum of $α$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_α(B)$ 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 $α$. In this paper, we prove that $g_α(B)\leq h_α(B)$, find upper and lower bounds for both…

照亮 ${\mathbb R}^3$ 和 ${\mathbb R}^4$ 中的 1-无条件凸体，以及更高维度中的某些情形

Illuminating 1-unconditional convex bodies in ${\mathbb R}^3$ and ${\mathbb R}^4$, and certain cases in higher dimensions

本文解决了三维和四维空间中所有1-无条件凸体的照明猜想，并推广到具有特定坐标投影的更高维情形，确认了立方体类子集的等式情形，方法基于组合技术和深照明概念。