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Background

Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence $(\lambda_n)_{n \in {\mathbb N}}$ of nonnegative real numbers such that $$\sum_{n = 1}^{\infty} \lambda_n = 1\,,$$ the series $$\sum_{n = 1}^{\infty} \lambda_nx_n$$ belongs to $A$.

Question

The statement that every nonempty, open convex set is ideally convex is presented as an exercise in about half a dozen books I've consulted so far. None of them seems to even outline a proof of this fact. The note by Lifshits in which the concept of ideally convex sets was introduced$^{2}$ implies that he was not aware of this "fact," since it only goes as far as stating that

the closure of a convex set is ideally convex,

and that

the simplest example of a non-closed ideally convex set is [an] open ball.

(These two do follow straight from the definition.) I've tried to prove this for a couple of weeks, now. My intuition is that it is true, but I can't put my finger on it. I cannot seem to be able to control the size of the tail of the series so as to guarantee that the limit is not in the boundary of the set. Does anybody know whether this is true? Could you outline the proof, or point me to a reference in which it is done? Thanks!

Footnotes

  1. Holmes (in Geometric Functional Analysis and its Applications, Springer-Verlag, 1975) defines the concept for general linear topological spaces, and seems to believe that open convex sets are still ideally convex in this general context. I'd be happy to see a proof for Banach spaces, but wouldn't mind seeing a more general proof, if that's available. (Pages 138--139.)

  2. My apologies for those who may not have access to the paper. I believe I included all you need to know from it in the quotations, though.

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The proof is not difficult (write the infinite convex combinations as a finite convex combination of nearby points...). Check also this question and comments mathoverflow.net/questions/56161/… –  Pietro Majer Jan 26 at 22:31
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