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Distance sets corresponding to convex bodies. (English) Zbl 1075.52004

Let \(A \subset \mathbb R^d\) be a measurable set with a positive upper density. It was proved by Furstenberg, Katznelson and Weiss, by Falconer and Marstrand (for \(d=2\)), and by Bourgain that the distance set \[ D(A):=\{\| x-y\| : x,y \in A \} \] (where \(\| \cdot \| \) is the Euclidean norm) contains \([t_0, \infty)\) for some \(t_0 > 0\).
The present paper concerns the following question: under what assumption on a convex body \(K\) symmetric with respect to 0, this result will remain valid if the Euclidean norm is replaced by the norm \(\| \cdot \| _K\) defined by \[ \| x\| _K:= \sup\{t\geq 0: x \notin tK \}. \] Let \[ D_K(A):= \{\| x-y\| _K : x,y \in A \}. \] The author proves that if \(K\) is not a polytope or is a polytope “with many faces”, then (as for \(K\) being the Euclidean ball) \(D_K(A)\) contains \([t_0, \infty)\) for some \(t_0 > 0\).

MSC:

52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
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