A characterization of all loglinear inequalities for three quermassintegrals of convex bodies

Peter Gritzmann

doi:10.1090/S0002-9939-1988-0962829-5

A characterization of all loglinear inequalities for three quermassintegrals of convex bodies

Peter Gritzmann

University of Trier

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

6 Scopus citations

Abstract

We giv;…omplete characterization of all inequalities of the type W_i^α(K)W_j^β(K)W_k^γ(K)≥ c, wher;…s an arbitrary convex body of Euclidean (d-space, W_l(K);… i, j, k, denotes the ith quermassintegral o;…nd α, β, γ an;…re arbitrary reals;…pecial case of such inequalities is the classical isoperimetric inequality for the volume and surface area of convex bodies. It turns out that all nontrivial inequalities of this type can be generated by meansof d–1 basic inequalities, the Fenchel-Alexandrov inequalities.

Original language	English
Pages (from-to)	563-570
Number of pages	8
Journal	Proceedings of the American Mathematical Society
Volume	104
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9939-1988-0962829-5
State	Published - 1 Oct 1988
Externally published	Yes

Keywords

Convex bodies
Fenchel-Alexandrov inequalities
Isoperimetric inequality
Quermassintegrals

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10.1090/S0002-9939-1988-0962829-5

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abstract = "We giv;…omplete characterization of all inequalities of the type Wiα(K)Wjβ(K)Wkγ(K)≥ c, wher;…s an arbitrary convex body of Euclidean (d-space, Wl(K);… i, j, k, denotes the ith quermassintegral o;…nd α, β, γ an;…re arbitrary reals;…pecial case of such inequalities is the classical isoperimetric inequality for the volume and surface area of convex bodies. It turns out that all nontrivial inequalities of this type can be generated by meansof d–1 basic inequalities, the Fenchel-Alexandrov inequalities.",

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AB - We giv;…omplete characterization of all inequalities of the type Wiα(K)Wjβ(K)Wkγ(K)≥ c, wher;…s an arbitrary convex body of Euclidean (d-space, Wl(K);… i, j, k, denotes the ith quermassintegral o;…nd α, β, γ an;…re arbitrary reals;…pecial case of such inequalities is the classical isoperimetric inequality for the volume and surface area of convex bodies. It turns out that all nontrivial inequalities of this type can be generated by meansof d–1 basic inequalities, the Fenchel-Alexandrov inequalities.

KW - Convex bodies

KW - Fenchel-Alexandrov inequalities

KW - Isoperimetric inequality

KW - Quermassintegrals

UR - https://www.scopus.com/pages/publications/84968516179

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DO - 10.1090/S0002-9939-1988-0962829-5

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SN - 0002-9939

VL - 104

SP - 563

EP - 570

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 2

ER -

A characterization of all loglinear inequalities for three quermassintegrals of convex bodies

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Keywords

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