On planar convex bodies of given Minkowskian thickness and least possible area

Abstract

Assume that the width of a planar convex body K at every direction u is not less than the width of a fixed centrally symmetric body B at the direction u. We present a precise description of those bodies K given above that have least possible area. Introduction.By E d , d 2, we denote the d-dimensional Euclidean space with origin o and norm | . |. Length and volume in E d are denoted by µ and V , respectively. The unit sphere in E d is denoted by S d−1 . A set K E d is said to be a convex body if it is convex, compact and has non-empty interior, cf.[4] and [21]. If u is ranging over E d , then the functions h K (u) := max{ x, u : x ∈ K} and w K (u) := h K (u)+h K (−u) are called the support function and the width function of K, respectively (cf.[16] and [9]). For u ∈ S d−1 , w K (u) is the width of K at direction u, i.e., the distance between the two different supporting hyperplanes of K being orthogonal to u. Let us fix an arbitrary convex body B E d centered at the origin. The object of our investigation is the class X (B) of convex bodies K E d satisfying the constraint w K (u) h B (u) for any u ∈ S d−1 and minimizing the volume. We introduce the Minkowskian thickness B (K) of a convex body K E d with respect to the gauge B by the equality B (K) := min{w K (u)/ h B (u) : u ∈ S d−1 }. If B is the Euclidean unit ball, then B (K) is the usual thickness (= minimal width). Obviously, we can describe X (B) as the class of convex bodies having Minkowskian thickness one and least possible volume. Furthermore, it is easy to see that the homothetical copies of bodies from X (B) correspond to the equality case in the geometric inequalityan arbitrary convex body and α(B) is the volume of convex bodies from X (B). The sharp estimates for the quantity α(B) are found in [1]. We cite the books [4, §10], [12, §6], [5], [6], and [22, Sections 4.4 and 4.5], where various geometric inequalities in Minkowski and Euclidean spaces are discussed. If d = 2 and B is the Euclidean unit disk, then the elements of X (B) are the planar convex bodies of Euclidean thickness one and least possible area. It was proved by Pál that for the above case X (B) is the class of equilateral

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