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Common tangents to convex bodies

It is well-known since the time of the Greeks that two disjoint circles in the plane have four common tangent lines. Cappell et al. proved a generalization of this fact for properly separated strictly convex bodies in higher dimensions. We have shown that the same generalization applies for arbitrary convex bodies. When the number of convex sets involved is equal to the dimension, we obtain an alternative combinatorial proof of Bisztriczky's theorem on the number of common tangents to $d$ separated convex bodies in $\Rr^d$.