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

TL;DR

This paper proves the Hadwiger-Boltyanski Illumination Conjecture for all 1-unconditional convex bodies in dimensions 3 and 4, and for certain higher-dimensional cases, using combinatorial methods and small perturbations of basis vectors.

Contribution

It establishes the conjecture for specific classes of 1-unconditional convex bodies in low and higher dimensions, extending previous partial results.

Findings

Confirmed the conjecture for all 1-unconditional bodies in R^3 and R^4.

Proved the conjecture for higher-dimensional bodies with certain projection properties.

Validated the conjecture's equality cases for bodies with no extreme points on coordinate subspaces.

Abstract

We settle the Hadwiger-Boltyanski Illumination Conjecture for all 1-unconditional convex bodies in ${\mathbb R}^3$ and in ${\mathbb R}^4$. Moreover, we settle the conjecture for those higher-dimensional 1-unconditional convex bodies which have at least one coordinate hyperplane projection equal to the corresponding projection of the circumscribing rectangular box. Finally, we confirm the conjectured equality cases of the Illumination Conjecture within the subclass of 1-unconditional bodies which, just like the cube $[-1,1]^n$, have no extreme points on coordinate subspaces. Our methods are combinatorial, and the illuminating sets that we use consist primarily of small perturbations of the standard basis vectors. In particular, we build on ideas and constructions from [Sun-Vritsiou, "On the illumination of 1-symmetric convex bodies", preprint available at arXiv:2407.10314], and mainly on the notion of 'deep illumination' introduced there.