Characterizing addition of convex sets by polynomiality of volume and by the homothety operation

Vitali Milman; Liran Rotem

doi:10.1142/S0219199714500229

Characterizing addition of convex sets by polynomiality of volume and by the homothety operation

Vitali Milman, Liran Rotem

School of Mathematical Sciences

Tel Aviv University

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

Abstract

We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

Original language	English
Article number	1450022
Journal	Communications in Contemporary Mathematics
Volume	17
Issue number	3
DOIs	https://doi.org/10.1142/S0219199714500229
State	Published - 25 Jun 2015

Keywords

Convex sets
Minkowski addition

All Science Journal Classification (ASJC) codes

Applied Mathematics
General Mathematics

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10.1142/S0219199714500229

Cite this

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title = "Characterizing addition of convex sets by polynomiality of volume and by the homothety operation",

abstract = "We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.",

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AU - Rotem, Liran

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N2 - We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

AB - We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

KW - Convex sets

KW - Minkowski addition

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U2 - 10.1142/S0219199714500229

DO - 10.1142/S0219199714500229

M3 - مقالة

SN - 0219-1997

VL - 17

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 3

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ER -

Characterizing addition of convex sets by polynomiality of volume and by the homothety operation

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

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