Journal of Convex Analysis 20 (2013), No. 2, 439--452
Copyright Heldermann Verlag 2013
Remarks on Diameter 2 Properties
Trond Abrahamsen
Dept. of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway
Trond.A.Abrahamsen@uia.no
Vegard Lima
Aalesund University College, Postboks 1517, 6025 Aalesund, Norway
Vegard.Lima@gmail.com
Olav Nygaard
Dept. of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway
Olav.Nygaard@uia.no
If X is an infinite-dimensional uniform algebra, if X has the Daugavet property or if X is a proper M-embedded space, every relatively weakly open subset of the unit ball of the Banach space X is known to have diameter 2, i.e., X has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming lp-sums of diameter 2 spaces does not ruin diameter 2 structure.
Keywords: Diameter 2, slice, Daugavet property, M-embedded, uniform algebra.
MSC: 46B20, 46B22
[ Fulltext-pdf (149 KB)] for subscribers only.