Convex sets with homothetic projections

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.

💡 Research Summary

The paper establishes a broad generalization of classical results by Suss and Hadwiger concerning homothetic relationships between convex bodies whose orthogonal projections onto lower‑dimensional subspaces are homothetic. The authors consider two convex sets (A) and (B) in Euclidean space (\mathbb{R}^n) (with (n\ge 3)). For a fixed integer (m) they examine every (m)-dimensional linear subspace (\Pi) and the orthogonal projection (\pi_\Pi) onto (\Pi). The central hypothesis is that for each such (\Pi) there exist a scalar (\lambda(\Pi)\neq0) (the homothety ratio) and a translation vector (t(\Pi)) such that

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