Reduced bodies in normed planes

Abstract

We say that a convex body R of a d-dimensional real normed linear space M d is reduced, if ∆(P ) < ∆(R) for every convex body P ⊂ R different from R. The symbol ∆(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M d . We establish a number of properties of reduced bodies in M 2 . They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M 2 is larger than ∆(R) for all directions strictly between two fixed directions and equals ∆(R) for these two directions.

Approximation of Spherical Bodies of Constant Width and Reduced Bodies

“…So we say that ⌢ gh and ⌢ g ′ h ′ is a pair of opposite curves of constant width ∆(R). Let us summarize the above consideration as the following claim analogous to Corollary 11 of [4] on the situation in a normed plane.…”