Common tangents to convex bodies

TL;DR

This paper extends classical results on common tangents from circles to arbitrary convex bodies in higher dimensions, providing new proofs and generalizations of existing theorems.

Contribution

The authors generalize the classical four-tangent theorem to arbitrary convex bodies and offer an alternative combinatorial proof of Bisztriczky's theorem.

Findings

Generalization of common tangent results to arbitrary convex bodies in any dimension

Proof of the number of common tangents for separated convex bodies in ^d

New combinatorial proof of Bisztriczky's theorem

Abstract

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 $\Rr^d$.