Complete sets need not be reduced in Minkowski spaces

Abstract

It is well known that in n-dimensional Euclidean space (n ≥ 2) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For n-dimensional Minkowski spaces, this coincidence is no longer true if n ≥ 3. Thus, the question occurs whether for n ≥ 3 any complete set is reduced. Answering this in the negative for n ≥ 3, we construct (2 k − 1)-dimensional (k ≥ 2) complete sets which are not reduced.Keywords bodies of constant width · (diametrically) complete sets · Hadamard matrix · Minkowski Geometry · reduced body · Walsh matrix Mathematics Subject Classification (2000) 46B20 · 52A20 · 52A21 · 52A40 · 52B11 IntroductionWe denote by X = (R n , · ) an n-dimensional normed or Minkowski space, i.e., an n-dimensional real Banach space with origin o. We will use the common abbreviations aff and conv for affine and convex hull, respectively. A convex body in X is a compact, convex set with interior points, and the norm · is induced by the o-symmetric convex body B X , called the unit ball of X, via x = min{λ ≥ 0 : x ∈ λB X }, ∀x ∈ R n .

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

“…[18]) that K cw ⊂ K cp and K cw = K cp in the planar case or for E n . Unfortunately, this is not the case in general Minkowski spaces when n ≥ 3 (as pointed out in [18] and [36]). For instance, let M n be a Minkowski space with indecomposable unit ball B, i. e. for all K, L ∈ K n with B = K + L, there exist λ, µ ≥ 0, s. t. λK = t µL = t B (cf.…”

Is a complete, reduced set necessarily of constant width?

“…As first shown in [17], the notion of reducedness can be carried over to Minkowski spaces, too; existing results and interesting research problems are presented in [19]. Since, in general, in such spaces complete bodies need not be reduced (see [22]) and one can construct reduced bodies which are not complete, the question is how these two classes are related to each other. Indeed, the family of bodies of constant width forms a subfamily of both.…”