On the geometry of locally nonconical convex sets (Q1290879)

If for every point \(x,y\) a convex subset \(Q\) of a Hausdorff topological vector space there exists a relative neighborhood \(U\) of \(x\) in \(Q\) such that \(U+{1\over 2}(y-x)\subset Q\), then we say that \(Q\) is locally nonconical. The author proves that a proper closed convex set \(Q\) with non-empty interior in a Hilbert space is locally nonconical if and only if for every line segment \([z,z'] \subset\text{bd}(Q)\) and for every \(p\in]z,z'[\), there exists a relative neighborhood \(U\) of \(p\) in \(Q\) such that the set \(U\cap \text{bd}(Q)\) is the union of a family of line segments parallel to \([z,z']\). It is shown that the closed unit ball of the Banach space of real sequences converging to 0 is locally nonconical.