Shadow systems and volumes of polar convex bodies

Mathieu Meyer; Shlomo Reisner

doi:10.1112/s0025579300000061

Shadow systems and volumes of polar convex bodies

Mathieu Meyer, Shlomo Reisner

Department of Mathematics

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

Abstract

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies K_t, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝ^d that have at most d + 3 vertices.

Original language	English
Pages (from-to)	129-148
Number of pages	20
Journal	Mathematika
Volume	53
Issue number	1
DOIs	https://doi.org/10.1112/s0025579300000061
State	Published - Jun 2006

Bibliographical note

Funding Information:
We have thus shown that b generates an extreme ray of C and that a ≡ 0. Hence, by Lemma 14, K is a triangle. h Acknowledgements. Both authors have been supported in part by the Research Network Program in Mathematics of the High Council for Scientific and Technological Cooperation between France and Israel. Additionally, Shlomo Reisner has been supported in part by NATO Collaborative Linkage Grant PST.CLG 979701.

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1112/s0025579300000061

Cite this

@article{2d1ee523493d4b65bf8f4ba6ba9a9de1,

title = "Shadow systems and volumes of polar convex bodies",

abstract = "It is proved that the reciprocal of the volume of the polar bodies, about the Santal{\'o} point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santal{\'o} inequality for polytopes in ℝd that have at most d + 3 vertices.",

author = "Mathieu Meyer and Shlomo Reisner",

note = "Funding Information: We have thus shown that b generates an extreme ray of C and that a ≡ 0. Hence, by Lemma 14, K is a triangle. h Acknowledgements. Both authors have been supported in part by the Research Network Program in Mathematics of the High Council for Scientific and Technological Cooperation between France and Israel. Additionally, Shlomo Reisner has been supported in part by NATO Collaborative Linkage Grant PST.CLG 979701.",

year = "2006",

month = jun,

doi = "10.1112/s0025579300000061",

language = "English",

volume = "53",

pages = "129--148",

journal = "Mathematika",

issn = "0025-5793",

publisher = "John Wiley and Sons Inc.",

number = "1",

}

TY - JOUR

T1 - Shadow systems and volumes of polar convex bodies

AU - Meyer, Mathieu

AU - Reisner, Shlomo

N1 - Funding Information: We have thus shown that b generates an extreme ray of C and that a ≡ 0. Hence, by Lemma 14, K is a triangle. h Acknowledgements. Both authors have been supported in part by the Research Network Program in Mathematics of the High Council for Scientific and Technological Cooperation between France and Israel. Additionally, Shlomo Reisner has been supported in part by NATO Collaborative Linkage Grant PST.CLG 979701.

PY - 2006/6

Y1 - 2006/6

N2 - It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

AB - It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

UR - http://www.scopus.com/inward/record.url?scp=33846888260&partnerID=8YFLogxK

U2 - 10.1112/s0025579300000061

DO - 10.1112/s0025579300000061

M3 - Article

AN - SCOPUS:33846888260

SN - 0025-5793

VL - 53

SP - 129

EP - 148

JO - Mathematika

JF - Mathematika

IS - 1

ER -

Shadow systems and volumes of polar convex bodies

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this