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Second-order convex functions were introduced by Mond (Opsearch 11(2-3):90-99, ) in order to deal with second-order duality. Then that notion was generalized again and again, using more and more parameters introduced using several quantifiers. In the present paper, we show that most of these notions have quite simple intrinsic characterizations. This paper can be viewed as a continuation of our paper (Zălinescu in An Ştiinţ Univ Al I Cuza Iaşi Secţ I a Mat 35(3):213-220, ) in which we characterized generalized bonvex functions. [ABSTRACT FROM AUTHOR]

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function. [ABSTRACT FROM PUBLISHER]

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach. [ABSTRACT FROM PUBLISHER]

The aim of this note is to show that the main results in Gao and Yang [Minimal distance between two non-convex surfaces, Optimization 57 (2008), pp. 705-714] are false and to discuss the manner in which the minimization problem in this article is investigated while presenting in a particular case a thorough study of the mentioned optimization problem. To achieve our main goal we provide counter-examples. [ABSTRACT FROM AUTHOR]