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- Title
The Kneser-Poulsen Conjecture for Special Contractions.
- Authors
Bezdek, Károly; Naszódi, Márton
- Abstract
The Kneser-Poulsen conjecture states that if the centers of a family of N unit balls in Ed are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that N≥(1+2)d. Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.
- Subjects
CONTRACTION operators; PAIRED comparisons (Mathematics); POLYHEDRA; LINEAR operators; EUCLIDEAN geometry
- Publication
Discrete & Computational Geometry, 2018, Vol 60, Issue 4, p967
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-018-9976-1