[Paper Review] Convex sets with homothetic projections

Valeriu Soltan|ArXiv.org|Mar 16, 2009

Point processes and geometric inequalities4 references4 citations

TL;DR

This paper establishes that compact or closed convex sets in ℝⁿ are homothetic if and only if their orthogonal projections onto every m-dimensional subspace (for 2 ≤ m ≤ n−1, or 3 ≤ m ≤ n−1 for unbounded sets) are homothetic, with the homothety ratio allowed to vary per projection. The proof relies on a refined version of Straszewicz’s theorem on exposed points and handles both positive and negative homothety, extending classical results by Süss and Hadwiger to the full homothety group.

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.

Motivation & Objective

• To extend Süss’s and Hadwiger’s theorems on positive homothety to full homothety (including negative ratios) in ℝⁿ.
• To determine the minimal dimension m of projection subspaces that guarantee homothety of compact or closed convex sets.
• To resolve the sharpness of the m ≥ 3 condition for unbounded convex sets by constructing a counterexample with 2D projections being homothetic but the sets not homothetic.
• To establish a relative version of the main theorem, showing homothety holds when projections are homothetic on all m-planes containing a fixed r-dimensional subspace.

Proposed method

• Use a refined version of Straszewicz’s theorem, which states that compact convex sets are the closed convex hull of their exposed points.
• Analyze the lineality and recession cones of convex sets to show that if projections on m-planes are homothetic, then the original sets must have the same lineality space.
• Reduce the problem to the case where both sets are line-free by decomposing them along their common lineality space.
• Use projection geometry in subspaces of dimension ≥ m ≥ 3 to show that homothety of projections on all m-planes containing a fixed line implies homothety of the original sets.
• Distinguish between positive and negative homothety in projections by analyzing recession cone symmetry and applying known results from [13] on homothety in higher dimensions.
• Apply a transitivity argument via intermediate projections: if projections on larger subspaces are homothetic, then projections on all lower-dimensional subspaces (e.g., 2D or 3D) are also homothetic, enabling application of Theorem 1.

Experimental results

Research questions

• RQ1 Under what conditions on m-dimensional projections are two compact or closed convex sets in ℝⁿ necessarily homothetic, including negative homothety?
• RQ2 Is the condition m ≥ 3 sharp for unbounded convex sets, or can m = 2 suffice?
• RQ3 Can the homothety of projections on all m-planes containing a fixed r-dimensional subspace imply global homothety of the original sets?
• RQ4 How does the behavior of exposed points and recession cones affect the continuity and structure of homothety ratios in projections?
• RQ5 To what extent can the classical Süss and Hadwiger theorems be generalized to include negative homothety ratios?

Key findings

• For compact convex sets in ℝⁿ, K₁ and K₂ are homothetic if and only if their orthogonal projections on every m-dimensional subspace (2 ≤ m ≤ n−1) are homothetic, with the homothety ratio allowed to depend on the projection plane.
• For closed (possibly unbounded) convex sets, the same equivalence holds provided m ≥ 3, showing that m = 2 is insufficient in the unbounded case.
• The example of two non-homothetic paraboloids in ℝ³ shows that 2-dimensional projections can be positively homothetic while the sets themselves are not homothetic, proving that m ≥ 3 is sharp for unbounded sets.
• The lineality spaces of K₁ and K₂ must coincide, as otherwise projection homothety would be violated on certain m-planes.
• When the sets are decomposed along their common lineality space, the problem reduces to the line-free case, where exposed point theory and projection geometry are applied.
• The relative version of the theorem (Corollary 1) shows that if projections on all m-planes containing a fixed r-dimensional subspace are homothetic, then the original sets are homothetic, extending the result to constrained projection families.