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Journal of Convex Analysis 18 (2011), No. 2, 301--339
Copyright Heldermann Verlag 2011



Tangency vis-Ó-vis Differentiability by Peano, Severi and Guareschi

Szymon Dolecki
Mathematical Institute of Burgundy, CNRS UMR 5584, Burgundy University, B. P. 47870, 21078 Dijon, France
dolecki@u-burgogne.fr

Gabriele H. Greco
Dip. di Matematica, UniversitÓ di Trento, 38050 Povo, Italy
greco@science.unitn.it



G. Peano defined differentiability of functions and lower tangent cones in 1887, and upper tangent cones in 1903, but uses the latter concept already in 1887 without giving a formal definition. Both cones were defined for arbitrary sets, as certain limits of appropriate homothetic relations. Around 1930 F. Severi and G. Guareschi, in a series of mutually fecundating individual papers, characterized differentiability in terms of lower tangent cones and strict differentiability in terms of lower paratangent cones, a notion introduced, independently, by Severi and G. Bouligand in 1928. Severi and Guareschi graduated about 1900 from the University of Turin, where Peano taught till his demise in 1932.

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