STABILITY OF AVERAGE ROUGHNESS, OCTAHEDRALITY, AND STRONG DIAMETER 2 PROPERTIES OF BANACH SPACES WITH RESPECT TO ABSOLUTE SUMS

We prove that, if Banach spaces X and Y are delta-average rough, then their direct sum with respect to an absolute norm N is delta/N (1; 1)-average rough. In particular, for octahedral X and Y and for p in (1; 1), the space X circle plus(p) Y is 2(1-1/p)-average rough, which is in general optimal. Another consequence is that for any delta in (1; 2] there is a Banach space which is exactly delta-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and infinity-sums.