Sections of simplices
We show that for ⌊d/2⌋≤k≤d, the relative interior of every k-face of a d-simplex Δd can be intersected by a 2(d−k)-dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results [2] show that the condition k≥⌊d/2⌋ above cannot be dropped and hence raise the question of determining, for al...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171299224015 |
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| Summary: | We show that for ⌊d/2⌋≤k≤d, the relative interior of every k-face of a d-simplex Δd can be intersected by a 2(d−k)-dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results [2] show that the condition k≥⌊d/2⌋ above cannot be dropped and hence raise the question of determining, for all 0≤k,j<d, an upper bound on the function c(j,k;d), defined as the smallest number of j-flats, j<d, needed to intersect the relative interiors of all the k-faces of Δd. Using probabilistic arguments, we show that C( j,k;d)≤(d+1k+1)(w+1k+1)log(d+1k+1), where w=min(max(⌊j2⌋+k,j),d). (*) |
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| ISSN: | 0161-1712 1687-0425 |