This is an announcement for the paper “On convex combinations of slices of the unit ball in Banach spaces” by Rainis Hallerhttps://arxiv.org/find/math/1/au:+Haller_R/0/1/0/all/0/1, Paavo Kuuseokhttps://arxiv.org/find/math/1/au:+Kuuseok_P/0/1/0/all/0/1, Märt Põldverehttps://arxiv.org/find/math/1/au:+Poldvere_M/0/1/0/all/0/1.
Abstract: We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the 1-sum of two Banach spaces does not have property (1), but it has property (2) if both the spaces have property (1); the Banach space $C[0,1]$ does not have property (2), although it has property (3).
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02919