[Paper Review] Hyperbolic $p$-sum and Horospherical $p$-Brunn-Minkowski theory in hyperbolic space

Haizhong Li, Botong Xu|arXiv (Cornell University)|Nov 13, 2022

Point processes and geometric inequalities4 citations

TL;DR

This paper introduces the hyperbolic p-sum in hyperbolic space H^{n+1}, establishing a novel horospherical p-Brunn-Minkowski theory for smooth, horospherically convex domains. By defining a new Minkowski-type sum and analyzing variations of modified quermassintegrals, the authors solve the horospherical p-Minkowski problem for all p ∈ (−∞, +∞) when the measure is even, and the horospherical p-Christoffel-Minkowski problem for p ∈ (−n, +∞) under suitable assumptions, proving key Brunn-Minkowski and Minkowski inequalities in the hyperbolic setting.

ABSTRACT

The classical Brunn-Minkowski theory studies the geometry of convex bodies in Euclidean space by use of the Minkowski sum. It originated from H. Brunn's thesis in 1887 and H. Minkowski's paper in 1903. Because there is no universally acknowledged definition of the sum of two sets in hyperbolic space, there has been no Brunn-Minkowski theory in hyperbolic space since 1903. In this paper, for any $p>0$ we introduce a sum of two sets in hyperbolic space, and we call it the hyperbolic $p$-sum. Then we develop a Brunn-Minkowski theory in the hyperbolic space by use of our hyperbolic $p$-sum, and we call it the horospherical $p$-Brunn-Minkowski theory. Let $K$ be any smooth horospherically convex bounded domain in the hyperbolic space $\mathbb{H}^{n+1}$. Through calculating the variation of the $k$-th modified quermassintegral of $K$ by use of our hyperbolic $p$-sum, we introduce the horospherical $k$-th $p$-surface area measure associated with $K$ on the unit sphere $\mathbb{S}^n$. For $k=0$, we introduce the horospherical $p$-Minkowski problem, which is the prescribed horospherical $p$-surface area measure problem. Through designing and studying a new volume preserving flow, we solve the existence of solutions to the horospherical $p$-Minkowski problem for all $p \in (-\infty,+\infty)$ when the given measure is even. For $1 \leq k \leq n-1$, we introduce the horospherical $p$-Christoffel-Minkowski problem, which is the prescribed horospherical $k$-th $p$-surface area measure problem. We solve the existence of solutions to the horospherical $p$-Christoffel-Minkowski problem for $p\in(-n, +\infty)$ under appropriate assumption on the given measure. We also study the Brunn-Minkowski inequalities and the Minkowski inequalities for domains in the hyperbolic space.

Motivation & Objective

• To establish a Brunn-Minkowski theory in hyperbolic space, which lacks a standard Minkowski sum due to non-Euclidean geometry.
• To define a new geometric sum, the hyperbolic p-sum, for sets in H^{n+1} using horospherical convexity and Minkowski norm properties.
• To formulate and solve the horospherical p-Minkowski problem for all p ∈ (−∞, +∞) when the measure is even.
• To extend the theory to the horospherical p-Christoffel-Minkowski problem for 1 ≤ k ≤ n−1 and p ∈ (−n, +∞), under appropriate measure conditions.
• To derive and prove hyperbolic p-Brunn-Minkowski and p-Minkowski inequalities for horospherically convex domains.

Proposed method

• Introduces the hyperbolic p-sum via a Minkowski-type construction in the hyperboloid model of H^{n+1}, using future time-like vectors and the Minkowski norm N.
• Defines the horospherical k-th p-surface area measure via variation of the k-th modified quermassintegral under the hyperbolic p-sum.
• Constructs a volume-preserving curvature flow to prove existence of solutions to the horospherical p-Minkowski problem.
• Uses concavity of the Minkowski norm N on future time-like vectors to prove key inequalities for 1 ≤ p ≤ 2.
• Applies the AM-GM inequality and geometric identities involving cosh distances to derive integral formulas for weighted volumes.
• Establishes the horospherical p-Brunn-Minkowski inequality by comparing the radius of a geodesic ball to the p-combination of radii under the hyperbolic p-sum.

Experimental results

Research questions

• RQ1 Can a consistent Minkowski-type sum be defined in hyperbolic space to support a Brunn-Minkowski theory?
• RQ2 Does the horospherical p-Minkowski problem admit solutions for all p ∈ (−∞, +∞) when the given measure is even?
• RQ3 Under what conditions does the horospherical p-Christoffel-Minkowski problem have solutions for p ∈ (−n, +∞)?
• RQ4 What are the hyperbolic analogues of the classical Brunn-Minkowski and Minkowski inequalities in the p-setting?
• RQ5 Can the origin-symmetry assumption in the hyperbolic p-Brunn-Minkowski conjecture be removed?

Key findings

• The horospherical p-Minkowski problem is solved for all p ∈ (−∞, +∞) when the given measure is even, using a volume-preserving curvature flow.
• The horospherical p-Christoffel-Minkowski problem is solved for p ∈ (−n, +∞) under appropriate assumptions on the given measure, extending the classical Christoffel-Minkowski problem to hyperbolic space.
• The hyperbolic p-Brunn-Minkowski inequality holds for horospherically convex domains, with equality if and only if the domains are horospherically congruent.
• For 1 ≤ p ≤ 2, the Minkowski norm N is concave on future time-like vectors, which is used to prove the p-Brunn-Minkowski inequality via convexity of the light cone.
• The weighted volume of geodesic balls not centered at the origin is derived using integral identities involving cosh distances, confirming the non-triviality of the p-sum construction.
• Counterexamples show that the origin-symmetry assumption in the hyperbolic p-Brunn-Minkowski conjecture cannot be removed, as the inequality fails when the center is far from the origin.