Convex sets with homothetic projections

TL;DR

This paper generalizes previous results by showing that convex sets in Euclidean space are homothetic if their projections on all m-dimensional planes are homothetic, using advanced convex geometry techniques.

Contribution

It extends known theorems to broader cases by proving convex sets are homothetic based on their projections, with a refined proof approach.

Findings

Convex sets are homothetic if all their m-dimensional projections are homothetic.

The homothety ratio may vary with the projection plane.

The proof employs a refined Straszewicz's theorem.

Abstract

Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given integer m between 2 and n - 1 (respectively, between 3 and n - 1), the orthogonal projections of the sets on every m-dimensional plane are homothetic, where homothety ratio and its sign may depend on the projection plane. The proof uses a refined version of Straszewicz's theorem on exposed points of compact convex sets.