We prove that, if Banach spaces and are -average rough, then their direct sum with respect to an absolute norm is -average rough. In particular, for octahedral and and for in , the space is -average rough, which is in general optimal. Another consequence is that for any in there is a Banach space which is exactly -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 -sums.
"Stability of average roughness, octahedrality, and strong diameter properties of Banach spaces with respect to absolute sums." Banach J. Math. Anal. 12 (1) 222 - 239, January 2018. https://doi.org/10.1215/17358787-2017-0040