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ON UNIFORM CONTRACTIONS OF BALLS IN MINKOWSKI SPACES
Author(s) -
Bezdek Károly
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12009
Subject(s) - mathematics , minkowski space , normed vector space , contraction (grammar) , unit sphere , combinatorics , minkowski addition , ball (mathematics) , euclidean space , euclidean geometry , mathematical analysis , geometry , medicine
Let N balls of the same radius be given in a d ‐dimensional real normed vector space, i.e., in a Minkowski d ‐space. Then apply a uniform contraction to the centers of the N balls without changing the common radius. Here 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. The main results of this paper state that a uniform contraction of the centers does not increase (respectively, decrease) the volume of the union (respectively, intersection) of N balls in Minkowski d ‐space, provided that N ≥ 2 d(respectively, N ≥ 3 dand the unit ball of the Minkowski d ‐space is a generating set). Some improvements are presented in Euclidean spaces.