Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$, then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$. We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$.

We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\text{AR}$($\sigma $-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on $X$.

We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.

There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of ℓ1. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:

1. 1) The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).

2. 2) The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).

3. 3) The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

Let {Pi} be a sequence of real (Laurent) polynomials each of which has no negative coefficients, and suppose that f is a real polynomial. Consider the problem of deciding whether

for all integers k, there exists Nsuch that the product of polynomials

(*) Pk+1. Pk+2.....Pk+N·ƒ has no negative coefficients.

In this paper we show that if X is a compact Hausdorff space, A is an arbitrary linear subspace of C(X, C), and if E is a Banach space, then each element L of (A ⊗ E)* can be represented by a boundary E*-valued vector measure of the same norm as L.