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Volume growth of Funk
geometry and the flags of polytopes
Dmitry Faifman, Constantin Vernicos and Cormac Walsh |
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Geometry & Topology 29 (2025) 3773–3811
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Abstract |
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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. |
Keywords
Funk metric, flag number, Mahler conjecture, $3^d$
conjecture, centro-affine surface area
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Mathematical Subject Classification
Primary: 51K99, 51M10, 52A38, 52A40, 52B05
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References |
Publication
Received: 8 December 2023
Revised: 24 October 2024
Accepted: 7 December 2024
Published: 10 October 2025
Proposed: Urs Lang
Seconded: Leonid Polterovich, Anna Wienhard
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Authors |
| Dmitry Faifman | |
| Département de Mathématiques et de
Statistique Université de Montréal Montréal, QC Canada |
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| Constantin Vernicos | |
| Institut Montpellierain Alexander
Grothendieck Université Montpellier Montpellier France |
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| Cormac Walsh | |
| Inria, CMAP, Ecole Polytechnique,
CNRS Université Paris-Saclay Palaiseau France |
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| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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