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Coconvex sets of finite volume Schneider, Rolf
Description
Let $C$ be a closed convex cone in $\mathbb{R}^n$, pointed and with interior points. A set $A = C \setminus K$, where $K \subset C$ is a closed convex set, is called a $C$-coconvex set if it has finite volume. The family of $C$-coconvex sets is closed under the addition $\oplus$ defined by $$C \setminus (A_1 \oplus A_2) = (C \setminus A_1) + (C \setminus A_2).$$ For compact $C$-coconvex sets, Khovanskii and Timorin (2014) have proved counterparts to the inequalities of the classical Brunn--Minkowski theory. For coconvex sets of finte volume, we prove a Brunn--Minkowski type inequality with equality discussion and then, as far as possible, counterparts to the uniqueness and existence theorems for sets with given surface area measures or cone-volume measures.
Item Metadata
Title |
Coconvex sets of finite volume
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-03-29T09:30
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Description |
Let $C$ be a closed convex cone in $\mathbb{R}^n$, pointed and
with interior points. A set $A = C \setminus K$, where $K \subset C$ is
a closed convex set, is called a $C$-coconvex
set if it has finite volume. The family of $C$-coconvex sets is closed under the
addition $\oplus$ defined by $$C \setminus (A_1 \oplus A_2) = (C \setminus A_1)
+ (C \setminus A_2).$$ For compact $C$-coconvex sets, Khovanskii and Timorin (2014)
have proved counterparts to the inequalities of the classical Brunn--Minkowski theory.
For coconvex sets of finte volume, we prove a Brunn--Minkowski type inequality with
equality discussion and then, as far as possible, counterparts to the uniqueness and
existence theorems for sets with given surface area measures or cone-volume
measures.
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Extent |
30 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Freiburg
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Series | |
Date Available |
2018-09-26
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0372155
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International