A number of useful inequalities sharpen the triangle inequality in $L^p$ spaces. The well known inequalities of Clarkson express the uniform convexity of the unit ball in $L^p$, $1 < p < \infty$, and later the sharp form of this inequality was found by Hanner. In 2006, Carbery conjectured  another such improved inequality  -- one that would be saturated not only when the pair of functions involved is linearly dependent, but also when they have disjoint supports. This talk presents joint work with Rurpert Frank, Paata Ivanisvilli and Elliott Lieb in which we prove the inequality conjectured by Carbery by actually proving a stronger version. We also prove a related inequality for sums on  an arbitrary finite number of functions in $L^p$. Finally, we prove some of these results for the Schatten trace norms on matrices, but here a number of questions remain open