Let K be a convex body in ? ~d. It is known that there is a constant C _0 depending only on d such that the probability that a random copy ρ(K) of, and this is best possible. We show that for every k < d there is a constant C such that the probability that ρ(K) contains a subset of dimension k is smaller than, This is best possible if k = d - 1. We conjecture that this is not best possible in the rest of the cases; if d = 2 and k = 0 then we can obtain better bounds. For d = 2, we find the best possible value of C _0 in the limit case when width(K)→0 and {pipe}K{pipe}~(→∞).
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