A new characterization of strict convexity on normed linear spaces

We consider relations between the distance of a set $A$ and the distance of its translated set $A+x$ from 0, for $x\in A$, in a normed linear space. If the relation d(0,A+x)<d(0,A)+\|x\| holds for exactly determined vectors $x\in A$, where $A$ is a convex, closed set with positive distance from 0, which we call (TP) property, then this property is equivalent to strict convexity of the space. We show that in uniformly convex spaces the considered property holds

ON KURATOWSKI MEASURE OF NONCOMPACTNESS IN R 2 WITH THE RIVER METRIC

In this paper we onsider some properties of the Kuratowski measure of noncompatness on the space (R2 , d* ), where d* is river metric. We prove the existence of the α-minimal sets in the given space, but also the strict minimalizability of the Kuratowski measure of noncompactness