Convex bodies with pairs of sections associated by reflections

TL;DR

This paper proves that under specific reflective symmetry conditions involving hyperplane sections and points outside the hyperplane, two convex bodies in higher-dimensional space are related by a reflection.

Contribution

It establishes a new geometric characterization linking local reflective symmetries of convex bodies to a global reflection mapping.

Findings

Existence of a reflection mapping between convex bodies under given conditions.

Local hyperplane section reflections imply a global reflection symmetry.

Results apply to convex bodies in dimensions three and higher.

Abstract

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M \subset H$, there exists a reflection mapping the hypersection of $K_1$ defined by $\mathrm{aff}\{p_1, M\}$ onto the hypersection of $K_2$ defined by $\mathrm{aff}\{p_2, M\}$, then there exists a reflection which maps $K_1$ onto $K_2$.