Given a convex subset $X$ of a real vector space $V$, I'm interested in the set $$Y:=\{x\in X:\ \forall v\in V, \ \exists\epsilon>0 \text{ s.t. } x+\epsilon v \in Y \}.$$

My question is boring: Does $Y$ have a standard name?

I would be tempted to call it the "algebraic interior" of $X$ or the "geometric interior" of $X$, but I don't recall hearing these names before. Given just this structure, I feel like this is what I would assume somebody means if they referred to the interior of $X$, but my application of interest also has a topology sitting around, and I don't want to refer to the interior under that topology.


1 Answer 1


This is indeed called the algebraic interior (and sometimes the radial kernel or core); although the last bit is usually formulated as "$x+tv\in Y$ for all $t\in [0,\epsilon]$" (which makes no difference for convex sets, of course).

It is one of the main lemmas of functional analysis (obtained as a fairly direct consequence of the Baire Category Theorem and used to show the Banach-Steinhaus Theorem) that for a closed and convex subset of a Banach space, the algebraic interior coincides with the usual topological interior.


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