منابع مشابه
Diametrically complete sets in Minkowski spaces
We obtain a new characterization of the diametrically complete sets in Minkowski spaces, by modifying two well-known characteristic properties of bodies of constant width. We also get sharp inequalities for the circumradius and inradius of a diametrically complete set of given diameter. Strengthening former work of D. Yost, we show that in a generic Minkowski space of dimension at least three t...
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If X is a Minkowski space, i.e. a finite dimensional real normed space, then S ⊂ X is an equilateral set if all pairs of points of S determine the same distance with respect to the norm. Kusner conjectured that e(`p) = d +1 for 1 < p < ∞ and e(`1) = 2d [6]. Using a technique combining linear algebra and approximation theory, we prove that for all 1 < p < ∞, there exists a constant Cp > 0 such t...
متن کاملCardinalities of k-distance sets in Minkowski spaces
A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1) points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is...
متن کاملMaximal Equilateral Sets
A subset of a normed space X is called equilateral if the distance between any two points is the same. Let m(X) be the smallest possible size of an equilateral subset of X maximal with respect to inclusion. We first observe that Petty’s construction of a d-dimensional X of any finite dimension d ≥ 4 with m(X) = 4 can be generalised to show that m(X⊕1 R) = 4 for any X of dimension at least 2 whi...
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In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1, . . . , n}, the number Dn is defined as the least number of points the space must contain in order to be sure that there will be an equilat...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1971
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1971-0275294-8