Some questions about rotundity and renormings in Banach spaces

In this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

There are no denting points in the unit ball of 𝒫(2H)

For any infinite dimensional real Hilbert space H we show that the unit ball of the space of continuous 2-homogeneous polynomials on H, 𝒫(2H), has no denting points. Thus the unit ball of 𝒫(2H) has no strongly exposed points.

Strongly exposed points in the unit ball of trace-class operators

A theorem of Arazy shows that every extreme point of the unit ball of trace-class operators is strongly exposed. We give this result a simpler and direct proof here.