The Santaló point for the Holmes-Thompson boundary area

Florent Balacheff; Gil Solanes; Kroum Tzanev

doi:https://doi.org/10.1016/j.aim.2021.108118

The Santaló point for the Holmes-Thompson boundary area

Florent Balacheff^*, Gil Solanes, Kroum Tzanev

^*Corresponding author for this work

Research output: Contribution to journal › Article › Research › peer-review

Abstract

We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is C¹, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santaló point expressed as an average of centroids of projections of the dual body.

Original language	English
Article number	108118
Journal	Advances in Mathematics
DOIs	https://doi.org/10.1016/j.aim.2021.108118
Publication status	Published - 1 Nov 2021

Keywords

Convex body
Crofton formula
Holmes-Thompson area and volume
Minkowski geometry
Santaló point
Symplectic geometry

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https://doi.org/10.1016/j.aim.2021.108118

https://ddd.uab.cat/record/250618

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title = "The Santal{\'o} point for the Holmes-Thompson boundary area",

abstract = "We explore the notion of Santal{\'o} point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is C1, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santal{\'o} point expressed as an average of centroids of projections of the dual body.",

keywords = "Convex body, Crofton formula, Holmes-Thompson area and volume, Minkowski geometry, Santal{\'o} point, Symplectic geometry",

author = "Florent Balacheff and Gil Solanes and Kroum Tzanev",

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AU - Solanes, Gil

AU - Tzanev, Kroum

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N2 - We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is C1, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santaló point expressed as an average of centroids of projections of the dual body.

AB - We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is C1, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santaló point expressed as an average of centroids of projections of the dual body.

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KW - Crofton formula

KW - Holmes-Thompson area and volume

KW - Minkowski geometry

KW - Santaló point

KW - Symplectic geometry

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JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 108118

ER -

The Santaló point for the Holmes-Thompson boundary area

Abstract

Keywords

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