Illuminating spiky balls and cap bodies
K\'aroly Bezdek, Ilya Ivanov, and Cameron Strachan

TL;DR
This paper establishes upper bounds on the illumination numbers of certain convex bodies called spiky balls and cap bodies, proving the Illumination Conjecture for high-dimensional centrally symmetric cases.
Contribution
It proves the Illumination Conjecture for high-dimensional centrally symmetric cap bodies and improves bounds for 1-unconditionally symmetric cases.
Findings
Centrally symmetric cap bodies in dimensions ≥20 can be illuminated by fewer than 2^d directions.
The paper confirms the Illumination Conjecture for these bodies in high dimensions.
Strengthens results for 1-unconditionally symmetric cap bodies.
Abstract
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 -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 -dimensional centrally symmetric cap body can be illuminated by directions in Euclidean -space for all . Furthermore, we strengthen the latter result for -unconditionally symmetric cap bodies.
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Taxonomy
TopicsPoint processes and geometric inequalities · Computational Geometry and Mesh Generation · Quasicrystal Structures and Properties
