On a functional equation related to a pair of hedgehogs with congruent projections

TL;DR

This paper proves that two hedgehogs in Euclidean space are congruent up to translation and reflection if their 2D projections are directly congruent without symmetries, based on a functional equation involving support functions.

Contribution

It establishes a new uniqueness result for hedgehogs based on their projections and a general functional equation approach, extending previous geometric knowledge.

Findings

Hedgehogs are uniquely determined by their 2D projections under certain conditions.

The result applies to hedgehogs in spaces of dimension three and higher.

A general analytic framework for solutions of a related functional equation is developed.

Abstract

Hedgehogs are geometrical objects that describe the Minkowski differences of arbitrary convex bodies in the Euclidean space $\mathbb{E}^n$. We prove that two hedgehogs in $\mathbb{E}^n, n \geq 3$, coincide up to a translation and a reflection in the origin, provided that their projections onto any two-dimensional plane are directly congruent and have no direct rigid motion symmetries. Our result is a consequence of a more general analytic statement about the solutions of a functional equation in which the support functions of hedgehogs are replaced with two arbitrary twice continuously differentiable functions on the unit sphere.