We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when consider...

This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of non-polyconvex but multiplicative quasiconvex isotropic functions. Relevance of...

The main aim of this paper is to introduce three classes $H^0_{p,q}$, $H^1_{p,q}$ and $TH^*_p$ of $p$-harmonic mappings and discuss the properties of mappings in these classes. First, we discuss the starlikeness and convexity of mappings in $H^0_{p,q}$ and $H^1_{p,q}$. Then establish the covering theorem for mappings in $H^1_{p,q}$. Finally, we determine the extreme points of the class $TH^*_{p}$.

The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank-one convexity or quasiconvexity. We give in theorem 6.6 simple necessary and sufficient conditions for an isotropic object...

The main result of this paper is a convexity theorem for momentum mappings J :M → g∗ of certain hamiltonian actions of noncompact semisimple Lie groups. The image of J is required to fall within a certain open subset D of g∗ which corresponds roughly via the orbit method to the discrete series of representations of the group G. In addition, J is required to be proper as a map from M to D. A rel...

The main result of this paper is a convexity theorem for momentum mappings J :M → g∗ of certain hamiltonian actions of noncompact semisimple Lie groups. The image of J is required to fall within a certain open subset D of g∗ which corresponds roughly via the orbit method to the discrete series of representations of the group G. In addition, J is required to be proper as a map from M to D. A rel...