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INTERSECTIONS OF BALLS AND SETS OF CONSTANT WIDTH IN FINITE‐DIMENSIONAL NORMED SPACES
Author(s) -
Martini Horst,
Richter Christian,
Spirova Margarita
Publication year - 2013
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/s0025579313000041
Subject(s) - mathematics , ball (mathematics) , normed vector space , banach space , intersection (aeronautics) , bounded function , combinatorics , constant (computer programming) , hull , norm (philosophy) , vector space , separable space , finite set , mathematical analysis , discrete mathematics , geometry , marine engineering , computer science , law , political science , engineering , programming language , aerospace engineering
For an arbitrary subset X of a finite‐dimensional real Banach space E , the ball intersection with parameter λ > 0 is defined as the intersection of all balls of radius λ whose centers are in X . On the other hand, the intersection of all balls of radius λ that contain X is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite‐dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non‐empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.