[Paper Review] On the shadow boundary of a centrally symmetric convex body

Ákos G. Horváth|ArXiv.org|Jun 20, 2007

Point processes and geometric inequalities14 references3 citations

TL;DR

This paper investigates the topological structure of the shadow boundary of a centrally symmetric convex body in n-dimensional Euclidean space, establishing that if the bisector (set of points equidistant to two points under a Minkowski norm) is a topological hyperplane, then the shadow boundary is homeomorphic to an (n−2)-sphere. The key contribution is proving that in the manifold case, such embeddings are standard, meaning the bisector is a topological hyperplane if and only if the shadow boundary is a topological (n−2)-sphere.

ABSTRACT

We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove that the shadow boundary is a topological manifold if all of the non-degenerated general parameter spheres are, too. In this case, using the approximation theorem of cell-like maps we get that they are homeomorphic to the $(n-2)$-dimensional sphere $S^{(n-2)}$. We also prove that the bisector (equidistant set of the corresponding normed space) in the direction ${\bf x}$ is homeomorphic to $R^{(n-1)}$ iff all of the non-degenerated general parameter spheres are $(n-2)$-manifolds implying that if the bisector is a homeomorphic copy of $R^{(n-1)}$ then the corresponding shadow boundary is a topological $(n-2)$-sphere.

Motivation & Objective

• To clarify the topological relationship between the shadow boundary of a centrally symmetric convex body and the structure of bisectors in Minkowski spaces.
• To determine whether the condition that all bisectors are topological hyperplanes implies strict convexity of the unit ball.
• To investigate whether the shadow boundary being a topological (n−2)-sphere characterizes the bisector as a topological hyperplane.
• To analyze the embedding properties of bisectors and shadow boundaries in the manifold case, particularly whether they admit non-standard (wild) embeddings.

Proposed method

• Define the shadow boundary S(K,x) as the set of boundary points of K where a line in direction x supports K without entering its interior.
• Use orthogonal projection px onto a hyperplane perpendicular to x to relate the shadow boundary to the boundary of the projection of K.
• Characterize the positive and negative parts K+ and K− of bdK as open, arc-wise connected, and homeomorphic to R^{n-1}.
• Introduce general parameter spheres γλ(K,x) as level sets of the gauge function, and study their topological properties via approximation by cell-like mappings.
• Apply M.Brown’s theorem on unions of increasing sequences of homeomorphic copies of R^{n-1} to prove that the bisector Hx is homeomorphic to R^{n-1} if all non-degenerate parameter spheres are homeomorphic to S^{n-2}.
• Use compactification and extension of homeomorphisms to show that in the manifold case, the embedding of the bisector, shadow boundary, and parameter spheres are standard (i.e., equivalent to standard embeddings via ambient homeomorphism).

Experimental results

Research questions

• RQ1 Is the shadow boundary of a centrally symmetric convex body homeomorphic to an (n−2)-sphere if and only if the corresponding bisector is a topological hyperplane?
• RQ2 Can a bisector be a topological hyperplane without the shadow boundary being a topological (n−2)-sphere?
• RQ3 In the manifold case, are the embeddings of the bisector, shadow boundary, and general parameter spheres always standard (i.e., equivalent to standard embeddings under ambient homeomorphism)?
• RQ4 Does the topological structure of the shadow boundary imply that the bisector is a topological hyperplane, and vice versa?
• RQ5 Under what conditions does the manifold property of the general parameter spheres imply that the bisector is homeomorphic to R^{n-1}?

Key findings

• The shadow boundary S(K,x) is a compact, connected, (n−2)-dimensional set that separates the boundary of K into three disjoint parts: S(K,x), K+, and K−.
• The sets K+ and K− are homeomorphic to R^{n-1}, and their union is open and arc-wise connected.
• If the bisector Hx is homeomorphic to R^{n-1}, then the non-degenerate general parameter spheres γλ(K,x) for λ > λ₀ are homeomorphic to S^{n-2}, and the shadow boundary S(K,x) is a topological (n−2)-sphere.
• In the manifold case, the embeddings of the bisector Hx, the shadow boundary S(K,x), and the general parameter spheres γλ(K,x) are all standard, meaning they are equivalent to standard embeddings under a homeomorphism of the ambient space.
• The bisector Hx is a topological hyperplane if and only if the shadow boundary S(K,x) is a topological (n−2)-sphere, under the assumption that all relevant sets are manifolds.
• The proof relies on M.Brown’s theorem and the extension of homeomorphisms from compactified sets, showing that the complement components' cell-like structure ensures standard embeddings.