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Keywords:
distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; $r$-boundary
distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; $r$-boundary
Summary:
If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\mathop {{\rm dist}}^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\mathop {{\rm dist}}^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.
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