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On the convergence of Minkowski sums to the convex hull Fradelizi, Matthieu
Description
Let us define, for a compact set $A \subset \R^n$, the Minkowski averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). $$ I shall show some monotonicity properties of $A(k)$ towards convexity when considering, the Hausdorff distance, the volume deficit and a non-convexity index of Schneider. For the volume deficit, the monotonicity holds in dimension 1 but fails for $n\ge 12$, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, a strong form of monotonicity holds. And for the Hausdorff distance, some monotonicity holds for $k$ large enough, depending of the ambient dimension and not on the set $A$. Based on a work in collaboration with Mokshay Madiman, Arnaud Marsiglietti and Artem Zvavitch.
Item Metadata
Title |
On the convergence of Minkowski sums to the convex hull
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-06-01T09:45
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Description |
Let us define, for a compact set $A \subset \R^n$, the Minkowski averages of $A$:
$$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). $$
I shall show some monotonicity properties of $A(k)$ towards convexity when considering, the Hausdorff distance, the volume deficit and a non-convexity index of Schneider.
For the volume deficit, the monotonicity holds in dimension 1 but fails for $n\ge 12$, thus disproving a conjecture of Bobkov, Madiman and Wang.
For Schneider's non-convexity index, a strong form of monotonicity holds. And for the Hausdorff distance, some monotonicity holds for $k$ large enough, depending of the ambient dimension and not on the set $A$.
Based on a work in collaboration with Mokshay Madiman, Arnaud Marsiglietti and Artem Zvavitch.
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Extent |
40 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Université Paris-Est Marne-la-Vallée
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Series | |
Date Available |
2017-11-29
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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 |
10.14288/1.0360764
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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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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International