A direct approach is used to establish both Ball and Barthe's reverse isoperimetric inequalities for the unit balls of subspaces of L p . This approach has the advantage that it completely settles all the open uniqueness questions for these inequalities.
"Volume Inequalities for Subspaces of L p ." J. Differential Geom. 68 (1) 159 - 184, September 2004. https://doi.org/10.4310/jdg/1102536713