Some Characterization of Locally Nonconical Convex Sets

A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x, y Q there exists an open neighbourhood U of x such that . A set Q is local cylindric (LC) if for x, y Q, x y, z (x, y) there exists an open neighbourhood U of z such that U Q (equivalently: bd(Q) U) is a union of open segments parallel to x, y]. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in 3], where the implication LNC LC was proved in general, while the inverse implication was proved in case of Hilbert spaces.