Zonoids whose polars are zonoids, cannot have proper faces other than vertices or facets. However, there exist non-smooth zonoids whose polars are zonoids. Examples in R 3 and R 4 are given.1991 Mathematics Subject Classification. 52A21.

AIn 1938, I. J. Schoenberg asked for which positive numbers p is the function exp(kRxR p ) positive definite, where the norm is taken from one of the spaces % n p , q 2. The solution of the problem was completed in 1991, by showing that for every p ? (0, 2], the function exp(kRxR p ) is not positive definite for the % n q norms with q 2 and n 3. We prove a similar result for a more general class of norms, which contains some Orlicz spaces and q-sums, and, in particular, present a simple proof of the answer to Schoenberg's original question. Some consequences concerning isometric embeddings in L p spaces for 0 p 2 are also discussed.

Abstract. Let f ( j, k, n) denote the expected number of j-faces of a random k-section of the n-cube. A formula for f (0, k, n) is presented, and, for j ≥ 1, a lower bound for f ( j, k, n) is derived, which implies a precise asymptotic formula for f (n − m, n − l, n) when 1 ≤ l < m are fixed integers and n → ∞.

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It is shown that for every n^3 there exists a centrally symmetric convex body whose generating distribution has a degree which is at least n À 2a2. This answers a question of Goodey and Weil.