We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in R m×n , min(m, n) ≤ 2, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.
Let P denote the well-known class of Caratheodory functions of the form p(z) = 1+ciz-i , z e A = {z ∈ ℂ: \z\ < 1}, with positive real part in the unit disc and let H(M) stand for the class of holomorphic functions commonly bounded by M in A. In 1992, J. Fuka and Z. J. Jakubowski began an investigation of families of mappings p ∈ P fulfilling certain additional boundary conditions on the unit circle T. At first, the authors examined the class P(B, b; α) of functions defined by conditions given by the upper limits for two disjoint open arcs of T. After that, such boundary conditions given, in particular, by the nontangential limits, were assumed for different subsets of the unit circle. In parallel, G. Adamczyk started to search for properties of families, contained in H(M) and satisfying certain similar conditions on T. The present article belongs to the above series of papers. In the first section we will consider subclasses of V of functions satisfying some inequalities on several arcs of T, whereas in Sections 2 and 3-families of mappings f ∈ H(M) with conditions given for measurable subsets of the unit circle T.