Width-integrals of convex bodies

The width-integrals (Breitenintegrale) as introduced by Blaschke are examined. They are shown to satisfy a cyclic inequality similar to that satisfied by the cross-sectional measures (Quermassintegrale). Other similarities and inequalities between the width-integrals and the cross-sectional measures are shown to exist. The setting for this paper is Euclidean n-space Rn. Compact convex sets with nonempty interiors are called convex bodies and the space of all convex bodies endowed with the Hausdorff topology is denoted by JKn. We denote the unit n-ball and the unit (n 1)-sphere by U and Q, respectively. For a convex body K, we use W.(K) to denote its cross-sectional measure (Quermassintegral) of index i. The n-dimensional volume of K is written as V(K). For a) 6 Q, bK(co) is defined to be half the width of K in the direction co. Two bodies K and L are said to have similar width if there exists a constant A > O such that bK(o) = A bL(co) for all co 6 U. For K EYK and p 6 int K, we use KP to denote the polar reciprocal of K with respect to the unit sphere centered at p. For reference see Bonnesen and Fenchel [21 and Hadwiger [5]. Width-integrals (Breitenintegrale) were first considered by Blaschke [1, p. 851 and later by Hadwiger [5, p. 266]. The width-integral of index i is defined by: Definition. B (K) = I fb Z(co dS(oJ) [i e R; K 6 K] where dS is the (n 1)-dimensional volume element on U. We note that our definition differs slightly from that of Blaschke in that we multiply by suitable constants to normalize the B .'s. In particular, we have BP(U) = cn for all i E R and Bn(K) = cn for all K E Kn,where cn denotes the volume of the unit n-ball. The width-integral of index i is a map, B. :K R. z n It is positive, continuous, homogeneous of degree n i and invariant under Received by the editors May 2, 1974 and, in revised form, June 28, 1974. AMS (MOS) subject classifications (1970). Primary 52A40, 52A20.