[Paper Review] A note and a short survey on supporting lines of compact convex sets in the plane

Gábor Czédli, László L. Stachó|arXiv (Cornell University)|Dec 5, 2016

Point processes and geometric inequalities23 references3 citations

TL;DR

This paper provides two rigorous, elementary proofs that supporting lines of compact convex sets in the plane can be continuously and slowly slide-turned along the boundary, establishing the rectifiability and continuous parameterization of the space of supporting lines. The key contribution is Theorem 3.1, which formalizes the intuitive idea that supporting lines can be moved smoothly around a convex shape, with applications to geometric topology and convex geometry.

ABSTRACT

After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is elementary on purpose.

Motivation & Objective

• To rigorously prove that supporting lines of compact convex sets in the plane can be continuously and slowly slide-turned along the boundary, addressing a widely assumed but underdemonstrated geometric intuition.
• To provide two elementary, accessible proofs of Theorem 3.1, ensuring broad applicability and referenceability in future research.
• To establish that the space of supporting lines, denoted Sli(H), forms a rectifiable Jordan curve, enabling continuous parameterization.
• To demonstrate the utility of Theorem 3.1 via Corollary 3.2, showing the existence of a full cycle of common supporting lines between two disjoint compact convex sets.

Proposed method

• The first proof uses a continuous parameterization of the boundary ∂H via a unit-speed curve, mapping each point to its unique supporting line with direction, and shows this map is continuous and rectifiable.
• The second proof constructs a homeomorphism f from the boundary ∂H to the space of supporting lines Sli(H), proving it is Lipschitz continuous in the small, ensuring smoothness and rectifiability.
• The proofs rely on geometric continuity, the rectifiability of ∂H (established via known results), and the use of angular and Euclidean distances in R^4 to control variation in direction and point location.
• Auxiliary constructions, such as the use of dual points P* and direction projections, allow control over the distance between supporting lines, enabling Lipschitz estimates.
• The parameterization of Sli(H) is achieved via a closed curve h(t) on a scaled unit circle L·C_unit, ensuring a full 360° turn over time t ∈ L·C_unit.
• The continuity of the distance function dist(ℓ(t), H₂) is used to track transitions between left/right positions of a second convex set H₂ relative to the moving line ℓ(t).

Experimental results

Research questions

• RQ1 Can the intuitive idea of continuously sliding and turning a supporting line around a compact convex set be rigorously proven?
• RQ2 Is the space of all supporting lines of a compact convex set in the plane a rectifiable Jordan curve?
• RQ3 Can the supporting line be parameterized continuously such that both the contact point and direction vary smoothly?
• RQ4 What happens to the relative position of a second disjoint compact convex set H₂ during a full continuous turn of the supporting line around H₁?
• RQ5 Does the existence of a full cycle of common supporting lines between two disjoint compact convex sets follow from continuity and rectifiability?

Key findings

• Theorem 3.1 establishes that the space of supporting lines Sli(H) of a compact convex set H in the plane is a rectifiable Jordan curve, enabling continuous and smooth parameterization.
• The paper provides two distinct, elementary proofs of Theorem 3.1, both showing that supporting lines can be slide-turned slowly and continuously around the boundary of H.
• The map f: ∂H → Sli(H), assigning each boundary point to its supporting line, is shown to be Lipschitz continuous in the small, ensuring local smoothness.
• Corollary 3.2 proves that for two disjoint compact convex sets H₁ and H₂, there exists a full cycle of common supporting lines, with the line transitioning from H₂ on the right to left and back, via continuity.
• The distance function dist(ℓ(t), H₂) is continuous in time t, ensuring that transitions between left and right positions of H₂ relative to the moving line occur at well-defined, isolated times.
• The proof confirms that the full turn of the supporting line around H₁ results in exactly four common supporting lines, pairwise disjoint, corresponding to the four transition points t₁ to t₄.