The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

Abstract

In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, e. g. concerning the Helly dimension or the Banach-Mazur distance.2010 Mathematics Subject Classification. Primary 52A20; Secondary 52A40, 52A21.

Large Signed Subset Sums

“…, ε d+1 u d+1 }) = S, thus meaning that R(S) = 1, where R(•) stands for the circumradius (see [17,Proposition 2.1]). Hence, if D(S) denotes the diameter of S, then by Jung's theorem [24] (see also [3,Lemma 3;16,(3)]) we have that…”

“…Hence there exist signs such that In particular we have that thus meaning that , where stands for the circumradius (see [17, Proposition 2.1]). Hence, if denotes the diameter of S , then by Jung's theorem [24] (see also [3, Lemma 3; 16, (3)]) we have that Since the diameter of S is attained between two vertices of S , this means that either for...…”

Is a complete, reduced set necessarily of constant width?

“…6] for the possibly most comprehensive overview), but amongst all, the one receiving most attention is the Minkowski asymmetry. In [5] it is shown how it naturally relates to complete and constant width sets in Minkowski spaces. Moreover, it has been repeatedly used to sharpen and strengthen geometric inequalities and related results, cf.…”

Minkowski concentricity and complete simplices

“…Denoting the n-dimensional Euclidean unit ball by B n 2 , Santaló (in case of n = 2) [25] and later Vrécica [29,Corollary 1] showed that r(K, B n 2 ) + R(K, B n 2 ) ≤ D(K, B n 2 ). The above inequality appears frequently in the literature (see e. g. [4] and [13]) and has been generalized to arbitrary Minkwoski spaces [11] (and even to general Banach spaces [11,22]), resulting in the concentricity inequality (the name pointing out that for the equality case it is necessary that every incenter is also a circumcenter -see [7,Lemma 2.4]): For all K, C ∈ K n with C (centrally) symmetric it holds (4) r(K, C) + R(K, C) ≤ D(K, C).…”

“…For the concentricity statement let us assume without loss of generality that C is Minkowski centered. It is shown in[7] that K is complete w. r. t. C − C implies that r(K, C − C)(C − C) ⊂ K − c ⊂ R(K, C − C)(C − C)for some Minkowski center c of K, thus K and C − C are Minkowski concentric. However, R(K,C) r(K,C) ≤ s(K)s(C) now implies by using Lemma 3.1 (e) again that r(K, C)C = 1 + 1 s(C) r(K, C − C)C ⊂ r(K, C − C)(C − C) ⊂ K − c ⊂ R(K, C − C)(C − C) ⊂ (1 + s(C))R(K, C − C)C = R(K, C)C, which shows K and C are Minkowski concentric.…”