Journal of Convex Analysis 18 (2011), No. 2, 301--339
Copyright Heldermann Verlag 2011
Tangency vis-Ó-vis Differentiability by Peano, Severi and Guareschi
Mathematical Institute of Burgundy, CNRS UMR 5584, Burgundy University, B. P. 47870, 21078 Dijon, France
Gabriele H. Greco
Dip. di Matematica, UniversitÓ di Trento, 38050 Povo, Italy
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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