On the X-ray number of almost smooth convex bodies and of convex bodies of constant width

TL;DR

This paper investigates the X-ray numbers of specific convex bodies, proving the X-ray and Illumination Conjectures for almost smooth bodies in any dimension and for bodies of constant width in dimensions 3 to 6.

Contribution

It provides proofs of the X-ray and Illumination Conjectures for classes of convex bodies, extending known results to new categories and dimensions.

Findings

Proof of the X-ray Conjecture for almost smooth convex bodies in all dimensions.

Proof of the Illumination Conjecture for convex bodies of constant width in dimensions 3 to 6.

Extension of conjecture validity to higher-dimensional convex bodies.

Abstract

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.