A characterization of duality through section/projection correspondence in the finite dimensional setting

Vitali D. Milman; Alexander Segal; Boaz A. Slomka

doi:https://doi.org/10.1016/j.jfa.2011.08.007

A characterization of duality through section/projection correspondence in the finite dimensional setting

Vitali D. Milman, Alexander Segal, Boaz A. Slomka

School of Mathematical Sciences

Tel Aviv University

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

Abstract

In this paper, we show that the well-known duality operation in the context of convex bodies in Rn is completely characterized by its property of interchanging sections with projections. Our results are compared to results by Böröczky-Schneider and Artstein-Milman, who showed that in many cases, the property of order reversing is sufficient to determine a duality operation, up to obvious linear modifications. In fact, we provide another result that recovers a known characterization of duality by the property of order reversing, and up to a mild condition, also a characterization of duality by interchanging sections by projections.

Original language	English
Pages (from-to)	3366-3389
Number of pages	24
Journal	Journal of Functional Analysis
Volume	261
Issue number	11
DOIs	https://doi.org/10.1016/j.jfa.2011.08.007
State	Published - 1 Dec 2011

Keywords

Duality
Polar convex sets
Section-projection correspondence

All Science Journal Classification (ASJC) codes

Analysis

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https://doi.org/10.1016/j.jfa.2011.08.007

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@article{cc71f1015be943a7b60c981037133324,

title = "A characterization of duality through section/projection correspondence in the finite dimensional setting",

abstract = "In this paper, we show that the well-known duality operation in the context of convex bodies in Rn is completely characterized by its property of interchanging sections with projections. Our results are compared to results by B{\"o}r{\"o}czky-Schneider and Artstein-Milman, who showed that in many cases, the property of order reversing is sufficient to determine a duality operation, up to obvious linear modifications. In fact, we provide another result that recovers a known characterization of duality by the property of order reversing, and up to a mild condition, also a characterization of duality by interchanging sections by projections.",

keywords = "Duality, Polar convex sets, Section-projection correspondence",

author = "Milman, {Vitali D.} and Alexander Segal and Slomka, {Boaz A.}",

note = "Funding Information: ✩ The first and second named authors were partially supported by the ISF grant No. 387/09 and the third named author was partially supported by the ISF grant No. 865/07. * Corresponding author. Fax: +972 77 4220101. E-mail addresses: milman@post.tau.ac.il (V.D. Milman), segalale@gmail.com (A. Segal), boazslom@post.tau.ac.il, boazslomka@gmail.com (B.A. Slomka).",

year = "2011",

month = dec,

day = "1",

doi = "https://doi.org/10.1016/j.jfa.2011.08.007",

language = "الإنجليزيّة",

volume = "261",

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journal = "Journal of Functional Analysis",

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T1 - A characterization of duality through section/projection correspondence in the finite dimensional setting

AU - Milman, Vitali D.

AU - Segal, Alexander

AU - Slomka, Boaz A.

N1 - Funding Information: ✩ The first and second named authors were partially supported by the ISF grant No. 387/09 and the third named author was partially supported by the ISF grant No. 865/07. * Corresponding author. Fax: +972 77 4220101. E-mail addresses: milman@post.tau.ac.il (V.D. Milman), segalale@gmail.com (A. Segal), boazslom@post.tau.ac.il, boazslomka@gmail.com (B.A. Slomka).

PY - 2011/12/1

Y1 - 2011/12/1

N2 - In this paper, we show that the well-known duality operation in the context of convex bodies in Rn is completely characterized by its property of interchanging sections with projections. Our results are compared to results by Böröczky-Schneider and Artstein-Milman, who showed that in many cases, the property of order reversing is sufficient to determine a duality operation, up to obvious linear modifications. In fact, we provide another result that recovers a known characterization of duality by the property of order reversing, and up to a mild condition, also a characterization of duality by interchanging sections by projections.

AB - In this paper, we show that the well-known duality operation in the context of convex bodies in Rn is completely characterized by its property of interchanging sections with projections. Our results are compared to results by Böröczky-Schneider and Artstein-Milman, who showed that in many cases, the property of order reversing is sufficient to determine a duality operation, up to obvious linear modifications. In fact, we provide another result that recovers a known characterization of duality by the property of order reversing, and up to a mild condition, also a characterization of duality by interchanging sections by projections.

KW - Duality

KW - Polar convex sets

KW - Section-projection correspondence

UR - http://www.scopus.com/inward/record.url?scp=80053276332&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.jfa.2011.08.007

DO - https://doi.org/10.1016/j.jfa.2011.08.007

M3 - مقالة

SN - 0022-1236

VL - 261

SP - 3366

EP - 3389

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 11

ER -

A characterization of duality through section/projection correspondence in the finite dimensional setting

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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