Abstract. In this paper we consider thickness and thinness of the unit sphere for infinite dimensional Banach spaces, as proposed by Whitley in [Wh]. We compute the thickness of several outer sums of Banach spaces and derive conclusions about the almost Daugavet property for those spaces. We also find estimates of the thickness of unit sphere in $L_p$ spaces, which allows to decide whether those spaces are almost Daugavet. In the last section we also present some results on the thinness of the unit sphere of some Banach spaces.

AMS Subject Classification (1991): 46B03, 46B25, 46B04

Keyword(s): thickness and thinness of a Banach space, almost Daugavet property

Received March 8, 2011, and in final form July 21, 2011. (Registered under 15/2011.)