ar X iv : m at h / 97 07 20 9 v 1 [ m at h . M G ] 5 J ul 1 99 7 The Santaló - regions of a convex body ∗

Motivated by the Blaschke-Santaló inequality, we define for a convex body K in Rn and for t ∈ R the Santaló-regions S(K,t) of K.We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||Kx0| is minimal (see for instance [Sch]). This unique x0 is called the Santaló-point of K. Moreover the Blaschke-Santaló inequality says that |K||Kx0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K x| v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santaló-region of K. Observe that it follows from the Blaschke-Santaló inequality that the Santalópoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. the paper was written while both authors stayed at MSRI supported by a grant from the National Science Foundation. MSC classification 52