Consider the L^p triangle inequality for functions, |f+g| \leq |f|+|g|, which is saturated when f=g, but which is poor when f and g have disjoint support. Carbery proposed a slightly more complicated inequality to take into account the orthogonality, or lack of it, ofthe two functions. With Eric Carlen and Rupert Frank it has now been proved. In fact, a much stronger version has been proved. Actually, Carbery was mainly interested in (non-commutative) matrices and traces instead of functions and integrals, so there is still much to be done.

A. Carbery, 'Almost-orthogonality in the Schatten-von Neumann classes',J. Operator Theory 62 (2009), 151158.