Covering the boundary of a convex body with its smaller homothetic copies.
Discrete Mathematics(2019)
摘要
For each positive integer m
and any convex body K, denote by γm(K) the smallest positive number γ so that the boundary of K can be covered by m translates of γK. It is proved that, for each positive integer m, γm(K) is Lipschitz continuous on the space of affine equivalence classes of n-dimensional convex bodies endowed with the Banach–Mazur metric. Exact values of γm(K) for particular choices of planar convex bodies K
and positive integers m are also obtained. Moreover, a general way to estimate γm(K) for centrally symmetric convex bodies is presented.
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关键词
Convex body,Covering,Hadwiger’s covering conjecture,Schäffer constant,Smaller homothetic copy
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