Geodesic
Subdifferentiability and superdifferentiability of distance functions
Matematičeskie zametki, Tome 61 (1997) no. 4, pp. 530-542

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We obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Dem'yanov–Rubinov sense) of the distance in an arbitrary norm from a point to a set for the finitedimensional case. The geometric structure of the subdifferential and the superdifferential is described. 
@article{MZM_1997_61_4_a4,
     author = {S. I. Dudov},
     title = {Subdifferentiability and superdifferentiability of distance functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {530--542},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_4_a4/}
}
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S. I. Dudov. Subdifferentiability and superdifferentiability of distance functions. Matematičeskie zametki, Tome 61 (1997) no. 4, pp. 530-542. http://geodesic.mathdoc.fr/item/MZM_1997_61_4_a4/