On the illumination of 1-symmetric convex bodies

TL;DR

This paper presents a new uniform approach to prove the illumination conjecture for all 1-symmetric convex bodies across all dimensions, confirming several longstanding conjectures and simplifying the illuminating sets needed.

Contribution

The authors introduce an alternative method that proves the Hadwiger-Boltyanski Illumination Conjecture for all dimensions for 1-symmetric convex bodies and confirms related conjectures using only pairs of opposite directions.

Findings

Confirmed the illumination conjecture for all dimensions in the class of 1-symmetric convex bodies.

Showed that illuminating sets can consist solely of pairs of opposite directions.

Validated the X-ray conjecture for all 1-symmetric convex bodies.

Abstract

In ["Illumination of convex bodies with many symmetries", Mathematika 63 (2017)], Tikhomirov verified the Hadwiger-Boltyanski Illumination Conjecture for the class of 1-symmetric convex bodies of sufficiently large dimension. We propose an alternative approach which allows us to settle the conjecture for this class in all dimensions in a uniform way. We also demonstrate that an alternative approach was indeed needed for the low dimensions. Finally, with this alternative method it is possible to solely use illuminating sets which consist of pairs of opposite directions; we thus also answer a question by Lassak, who has suggested this may be possible for any origin-symmetric convex body. As a consequence of this, we can also confirm the X-ray conjecture by Bezdek and Zamfirescu for all 1-symmetric convex bodies.