[Paper Review] Uniqueness of radial centers of parallel bodies

Jun O’Hara|arXiv (Cornell University)|Sep 23, 2011

Point processes and geometric inequalities8 references4 citations

TL;DR

This paper establishes the uniqueness of radial centers of order $\alpha$ for parallel bodies of convex bodies in $\mathbb{R}^m$ when the offset distance $\delta$ exceeds a dimension-dependent constant multiple of the body's diameter. Using the moving plane method and boundary integral expressions for second derivatives of Riesz potentials, the authors prove that for $\delta \geq \varphi(m) \cdot \textrm{diam}(\widetilde{\varOmega})$, the $r^{\alpha-m}$-center is unique for all $\alpha > 0$, resolving a conjecture in specific geometric regimes.

ABSTRACT

We show the uniqueness of the radial centers of any order $α$ of a parallel body of a convex body $Ω$ in $\mathbb R^m$ at distance $δ$ if $δ$ is greater than the diameter of $Ω$ multiplied by a constant which depends only on the dimension $m$.

Motivation & Objective

• To resolve the conjecture that convex bodies have unique $r^{\alpha-m}$-centers for all $\alpha > 0$ under certain geometric conditions.
• To establish conditions under which the radial center of order $\alpha$ is unique for parallel bodies of convex bodies.
• To extend previous uniqueness results, which were limited to $\alpha \geq m+1$ or $\alpha \leq 1$, to all $\alpha > 0$.
• To provide a quantitative threshold on the offset distance $\delta$ relative to the diameter of the original body for uniqueness to hold.

Proposed method

• Applies the moving plane method to show that any $r^{\alpha-m}$-center of a parallel body must lie within the original convex body $\widetilde{\varOmega}$.
• Uses boundary integral representations of the second derivatives of the Riesz potential $V_{\varOmega}^{(\alpha)}(x)$ to analyze convexity/concavity properties.
• Derives explicit integral expressions for $\frac{\partial^2 V_{\varOmega}^{(\alpha)}}{\partial x_j^2}$ using the unit outer normal and surface measure on $\partial\varOmega$, distinguishing cases for $x \in \varOmega^c$ and $x \in \varOmega^\circ$.
• Introduces a cone-based comparison argument to estimate the sign of the second derivative, leveraging the geometry of the parallel body $\widetilde{\varOmega} + \delta B^m$.
• Defines a function $\varphi(m)$ such that for $\delta \geq \varphi(m) \cdot \textrm{diam}(\widetilde{\varOmega})$, the potential $V_{\widetilde{\varOmega}+\delta B^m}^{(\alpha)}$ is strictly convex or concave depending on $\alpha$, ensuring uniqueness of the extremum.
• Employs renormalization techniques and continuity arguments to extend results to the full range of $\alpha > 0$, including $\alpha \leq 1$.

Experimental results

Research questions

• RQ1 Under what conditions on the offset distance $\delta$ is the radial center of order $\alpha$ unique for a parallel body of a convex body in $\mathbb{R}^m$?
• RQ2 Can the uniqueness of $r^{\alpha-m}$-centers be established for all $\alpha > 0$, including $1 < \alpha < m+1$, beyond previously known ranges?
• RQ3 Is there a dimension-dependent threshold $\varphi(m)$ such that $\delta \geq \varphi(m) \cdot \textrm{diam}(\widetilde{\varOmega})$ guarantees uniqueness of the radial center?
• RQ4 How does the geometry of the parallel body $\widetilde{\varOmega} + \delta B^m$ influence the convexity/concavity of the Riesz potential $V_{\varOmega}^{(\alpha)}$?
• RQ5 Can the second derivative of the Riesz potential be analyzed via boundary integrals to infer extremum uniqueness?

Key findings

• For any convex body $\widetilde{\varOmega}$ with piecewise $C^1$ boundary in $\mathbb{R}^m$, if $\delta \geq \varphi(m) \cdot \textrm{diam}(\widetilde{\varOmega})$, then the $r^{\alpha-m}$-center of the parallel body $\widetilde{\varOmega} + \delta B^m$ is unique for all $\alpha > 0$.
• The function $\varphi(m)$ is explicitly defined as $\max\left\{10, \max_{1 \leq \alpha \leq \alpha_0(10)} f(\alpha)\right\}$, where $f(\alpha)$ arises from a geometric integral estimate in Lemma 3.7.
• For $m = 2$, the threshold is $\varphi(2) = \sqrt{3}$, ensuring uniqueness when $\delta \geq \sqrt{3} \cdot \textrm{diam}(\widetilde{\varOmega})$.
• For $m \geq 3$, the threshold $\varphi(m)$ exists and is finite, relying on the existence of $\alpha_0(b)$ such that for $\delta \geq b \cdot \textrm{diam}(\widetilde{\varOmega})$, uniqueness holds for $\alpha \in [\alpha_0, m+1)$.
• The second derivative of $V_{\widetilde{\varOmega}+\delta B^m}^{(\alpha)}$ is shown to be negative for $\alpha \leq 1$, confirming strict concavity and thus uniqueness of the maximum point.
• The proof relies on sign analysis of boundary integrals and cone-based comparisons, showing that the second derivative maintains a definite sign under the given $\delta$-threshold, ensuring extremum uniqueness.