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On non-central sections of the simplex, the cube and the cross-polytope Koenig, Hermann
Description
We determine the non-central hyperplane sections of the $n$-simplex of maximal volume which have a fixed large distance to the centroid - large in the sense that the distance is bigger than the distance of the centroid to the midpoint if the edges. This complements similar results of Moody, Stone, Zach and Zvavitch for the $n$-cube and of Liu and Tkocz for the $n$-cross-polytope. We also show that parallels to the extremal hyperplanes for the $n$-simplex, the $n$-cube and the $n$-cross-polytope still provide at least local maxima for smaller distances, in a specified distance range and for sufficiently large dimensions (e.g. $n \ge 10$). Moreover, we find the maximal perimeters of non-central hyperplane sections of these bodies with large distances to the center. By perimeter we mean the $(n-2)$-dimensional intersection of the hyperplane with the boundary of the convex body.
Item Metadata
Title |
On non-central sections of the simplex, the cube and the cross-polytope
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2020-02-10T11:16
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Description |
We determine the non-central hyperplane sections of the $n$-simplex of maximal volume which have a fixed
large distance to the centroid - large in the sense that the distance is bigger than the distance of the
centroid to the midpoint if the edges. This complements similar results of Moody, Stone, Zach and Zvavitch
for the $n$-cube and of Liu and Tkocz for the $n$-cross-polytope. We also show that parallels to the extremal
hyperplanes for the $n$-simplex, the $n$-cube and the $n$-cross-polytope still provide at least local maxima
for smaller distances, in a specified distance range and for sufficiently large dimensions (e.g. $n \ge 10$).
Moreover, we find the maximal perimeters of non-central hyperplane sections of these bodies with large distances
to the center. By perimeter we mean the $(n-2)$-dimensional intersection of the hyperplane with the boundary
of the convex body.
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Extent |
27.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Universitaet Kiel
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Series | |
Date Available |
2020-08-09
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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 |
10.14288/1.0392667
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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 Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International