The second order subdifferential and the Dupin indicatrices of a nondifferentiable convex function. (English) Zbl 0632.53009

Hiriart-Urruty, J.-B.; Seeger, A.

Proc. Lond. Math. Soc., III. Ser. 58, No. 2, 351-365 (1989).

We introduce the notions of Dupin indicatrices for “nonsmooth” convex surfaces in \({\mathbb{R}}^{n+1}\) which are graphs of convex functions defined on \({\mathbb{R}}^ n\). We study them in connection with the concept of second order subdifferential of convex functions, such as introduced and developed recently by the authors. Finally, second order subdifferentials are viewed as limit sets of difference quotients involving approximate subdifferentials, which sheds a new light on the “second order information” contained in these approximate subdifferentials.

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
26B25	Convexity of real functions of several variables, generalizations

Keywords:

Dupin indicatrices; convex surfaces; convex functions; second order subdifferential; approximate subdifferentials; nonsmooth functions