On Godbersen’s conjecture

Shiri Artstein-Avidan; Keshet Einhorn; Dan I. Florentin; Yaron Ostrover

doi:10.1007/s10711-015-0060-1

On Godbersen’s conjecture

Shiri Artstein-Avidan, Keshet Einhorn, Dan I. Florentin, Yaron Ostrover

School of Mathematical Sciences

Tel Aviv University

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

Abstract

We provide a natural generalization of a geometric conjecture of Fáry and Rédei regarding the volume of the convex hull of K ⊂ Rⁿ, and its negative image -K. We show that it implies Godbersen’s conjecture regarding the mixed volumes of the convex bodies K and -K. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes V(K[j], -K[n - j]), which is not far from Godbersen’s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti’s difference function inequality.

Original language	English
Pages (from-to)	337-350
Number of pages	14
Journal	Geometriae Dedicata
Volume	178
Issue number	1
DOIs	https://doi.org/10.1007/s10711-015-0060-1
State	Published - 24 Oct 2015

Keywords

Convex body
Convex hull
Difference body
Godbersen’s conjecture
Mixed volume
Rogers–Shephard inequality

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1007/s10711-015-0060-1

Cite this

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title = "On Godbersen{\textquoteright}s conjecture",

abstract = "We provide a natural generalization of a geometric conjecture of F{\'a}ry and R{\'e}dei regarding the volume of the convex hull of K ⊂ Rn, and its negative image -K. We show that it implies Godbersen{\textquoteright}s conjecture regarding the mixed volumes of the convex bodies K and -K. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes V(K[j], -K[n - j]), which is not far from Godbersen{\textquoteright}s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti{\textquoteright}s difference function inequality.",

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author = "Shiri Artstein-Avidan and Keshet Einhorn and Florentin, \{Dan I.\} and Yaron Ostrover",

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AU - Einhorn, Keshet

AU - Florentin, Dan I.

AU - Ostrover, Yaron

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N2 - We provide a natural generalization of a geometric conjecture of Fáry and Rédei regarding the volume of the convex hull of K ⊂ Rn, and its negative image -K. We show that it implies Godbersen’s conjecture regarding the mixed volumes of the convex bodies K and -K. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes V(K[j], -K[n - j]), which is not far from Godbersen’s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti’s difference function inequality.

AB - We provide a natural generalization of a geometric conjecture of Fáry and Rédei regarding the volume of the convex hull of K ⊂ Rn, and its negative image -K. We show that it implies Godbersen’s conjecture regarding the mixed volumes of the convex bodies K and -K. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes V(K[j], -K[n - j]), which is not far from Godbersen’s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti’s difference function inequality.

KW - Convex body

KW - Convex hull

KW - Difference body

KW - Godbersen’s conjecture

KW - Mixed volume

KW - Rogers–Shephard inequality

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DO - 10.1007/s10711-015-0060-1

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SP - 337

EP - 350

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -

On Godbersen’s conjecture

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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