Suppose that K is an n-dimensional convex body of volume one. The isodiametric inequality states that K contains a line segment whose length has the order of magnitude of sqrt{n}. Does there exist a subset A of K of volume 1/2, such that for any line in R^n, its intersection with A is of measure much smaller than sqrt{n}?
We show that in the case of L_p-balls of volume one in R^n, the sqrt{n} can be replaced by n^{ (p-2) / (4 p+2)} for p > 2, and this is tight. For 1 < p < 2, the behavior is different, and the dependence is logarithmic in the dimension. For p = 1 or p = infinity the answer turns out to be n^{1/4}, which is also the typical behavior in the case of product measures, where a certain universality property is observed. Joint work with D. Elboim.
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