Shell, Glenn C. On the geometry of locally nonconical convex sets. (English) Zbl 0937.52002 Geom. Dedicata 75, No. 2, 187-198 (1999). 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. Reviewer: Marek Lassak (Berlin) Cited in 1 ReviewCited in 1 Document MSC: 52A05 Convex sets without dimension restrictions (aspects of convex geometry) 46A55 Convex sets in topological linear spaces; Choquet theory 52A07 Convex sets in topological vector spaces (aspects of convex geometry) Keywords:locally noncompact set; stable convex set; Hilbert space PDFBibTeX XMLCite \textit{G. C. Shell}, Geom. Dedicata 75, No. 2, 187--198 (1999; Zbl 0937.52002) Full Text: DOI