Geodesic
Sufficient conditions of the metric function of differentiability
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 1 (1992), pp. 84-89

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We give the characterization of Banach spaces in wich for every nonempty closed set $M$ a metric function (=distance function to $M$) is Frechet differentiable at points of uniqueness and weakly approximative compactness. 
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     title = {Sufficient conditions of the metric function of differentiability},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {84--89},
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V. S. Balaganskii. Sufficient conditions of the metric function of differentiability. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 1 (1992), pp. 84-89. http://geodesic.mathdoc.fr/item/TIMM_1992_1_a5/