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GEOMETRY AND VOLUME PRODUCT OF FINITE DIMENSIONAL LIPSCHITZ-FREE SPACES

Abstract : The goal of this paper is to study geometric and extremal properties of the convex body B F (M) , which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space M. We investigate 1 and ∞-sums, in particular we characterize the metric spaces such that B F (M) is a Hanner polytope. We also characterize the finite metric spaces whose Lipschitz-free spaces are isometric. We discuss the extreme properties of the volume product P(M) = |B F (M) | · |B • F (M) |, when the number of elements of M is fixed. We show that if P(M) is maximal among all the metric spaces with the same number of points, then all triangle inequalities in M are strict and B F (M) is simplicial. We also focus on the metric spaces minimizing P(M), and on Mahler's conjecture for this class of convex bodies.
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https://hal-upec-upem.archives-ouvertes.fr/hal-02924433
Contributor : Matthieu Fradelizi <>
Submitted on : Friday, August 28, 2020 - 9:40:05 AM
Last modification on : Wednesday, September 9, 2020 - 2:34:01 PM

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  • HAL Id : hal-02924433, version 1

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Matthew Alexander, Matthieu Fradelizi, Luis García-Lirola, Artem Zvavitch. GEOMETRY AND VOLUME PRODUCT OF FINITE DIMENSIONAL LIPSCHITZ-FREE SPACES. 2020. ⟨hal-02924433⟩

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