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Volume growth of Funk geometry and the flags of polytopes

Dmitry Faifman, Constantin Vernicos, Cormac Walsh

TL;DR

This work analyzes the Holmes–Thompson volume of Funk metric balls inside convex bodies, linking volume growth to the underlying polytope’s combinatorics via flag decompositions. It proves that for unconditional bodies, Vol$_K(B_K(R))$ is minimized by Hanner polytopes for all radii, aligning with a dynamic version of Mahler’s conjecture and Kalai’s flag conjecture; equality cases are characterized. The authors derive the two highest-order terms of the volume growth for polytopal Funk geometries, showing the leading term depends only on the number of flags, while the second term encodes geometry and centers it via a unique Funk–Santaló point $s_ abla(P)$. In 2D, stationary points of the second-term functional are affinely equivalent to regular polygons, with the regular polygon maximizing the second-term measure at the optimal center. The paper also treats simplices with recursive volume formulas and offers several conjectures and questions bridging Mahler-type inequalities, flag theory, and centro-affine geometry in the Funk setting.

Abstract

We consider the Holmes--Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santaló point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon.

Volume growth of Funk geometry and the flags of polytopes

TL;DR

This work analyzes the Holmes–Thompson volume of Funk metric balls inside convex bodies, linking volume growth to the underlying polytope’s combinatorics via flag decompositions. It proves that for unconditional bodies, Vol is minimized by Hanner polytopes for all radii, aligning with a dynamic version of Mahler’s conjecture and Kalai’s flag conjecture; equality cases are characterized. The authors derive the two highest-order terms of the volume growth for polytopal Funk geometries, showing the leading term depends only on the number of flags, while the second term encodes geometry and centers it via a unique Funk–Santaló point . In 2D, stationary points of the second-term functional are affinely equivalent to regular polygons, with the regular polygon maximizing the second-term measure at the optimal center. The paper also treats simplices with recursive volume formulas and offers several conjectures and questions bridging Mahler-type inequalities, flag theory, and centro-affine geometry in the Funk setting.

Abstract

We consider the Holmes--Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santaló point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon.
Paper Structure (17 sections, 34 theorems, 184 equations, 3 figures)

This paper contains 17 sections, 34 theorems, 184 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Preliminaries
  4. Funk geometry
  5. Faces and Flags
  6. Taking the polar with respect to different points
  7. Properties of the Holmes--Thompson volume in Funk geometry
  8. The Mahler inequality for unconditional bodies
  9. A decomposition of the volume
  10. Combining a flag simplex and its own dual
  11. Combining a flag simplex with the dual of a neighbour
  12. Combining a flag simplex with the dual of a non-neighbour
  13. The contribution of a flag simplex
  14. The limit of the Funk--Santaló point for polytopal Funk geometries as the radius tends to infinity
  15. Stationary points of the 2nd-highest-order-term functional in 2-d
  16. ...and 2 more sections

Key Result

Theorem 1.2

For any polytope $P$ of dimension $n$,

Figures (3)

  • Figure 1: The integral of $1/xy$ over the infinite grey region is finite.
  • Figure 2: The contribution of a single flag simplex
  • Figure 3: The contribution of a flag simplex after the sum is rearranged

Theorems & Definitions (63)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Proposition 3.1
  • ...and 53 more