Examples of non-quasiconvex functions that are rank-one convex are rare. In this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials, developed to identify G-closed sets of positive codimensions. The algebraic methods are used to construct a set of materials of positive codimension that is close...

Recently, Hermite-Hadamard-type inequalities and their applications have attracted considerable interest, as shown in the book [1], for example. These inequalities have been studied for various classes of functions such as convex functions [1], quasiconvex functions [2–4], p-functions [3, 5], Godnova-Levin type functions [5], r-convex functions [6], increasing convex-along-rays functions [7], a...

In this note we construct new examples of quasiconvex functions defined on the set Sn×n of symmetric matrices. They are built on the k-th elementary symmetric function of the eigenvalues, k = 1, 2, ..., n. The idea is motivated by Šverák’s paper [S]. The proof of our result relies on the theory of the so-called k-Hessian equations, which have been intensively studied recently, see [CNS], [T], [...

<p style='text-indent:20px;'>We study convex and quasiconvex functions on a metric graph. Given set of points in the graph, we consider largest function below prescribed datum. We characterize this as unique viscosity subsolution to simple differential equation, <inline-formula><tex-math id="M1">\begin{document}$ u'' = 0 $\end{document}</tex-math></inline-formula> ...

A number of rules for the calculus of subdifferentials of generalized convex functions are displayed. The subdifferentials we use are among the most significant for this class of functions, in particular for quasiconvex functions: we treat the Greenberg-Pierskalla’s subdifferential and its relatives and the Plastria’s lower subdifferential. We also deal with a recently introduced subdifferentia...

A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator L0(Du,Du) = min{v ·D2u vT | |v| = 1, |v ·Du| = 0}. In two dimensions this is the mean curvature operator, and in any dimension L0(Du,Du)/|Du| is the first principal curvature of the surface S = u−1(c). Our main results include a comparison principle for L0(Du,Du) = g when g ...

Let Y be a subspace of topological vector space X, and A⊂X an open convex set that intersects Y. We say the property (QE) [property (CE)] holds if every continuous quasiconvex [continuous convex] function on A∩Y admits extension defined A. study relations between (CE) properties, proving always implies that, under suitable hypotheses (satisfied for example X is normed closed X), two properties ...