Geodesic
Properties of distance functions on convex surfaces and applications
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 247-269.

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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.
DOI : 10.1007/s10587-011-0010-5
Classification : 52A20, 53C45
Keywords: distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; $r$-boundary 
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Rataj, Jan; Zajíček, Luděk. Properties of distance functions on convex surfaces and applications. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 247-269. doi : 10.1007/s10587-011-0010-5. http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0010-5/

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