In 2006 Carbery raised a question about an improvement on the naïve norm inequality 
(||f+g||_p)^p ≤ 2^(p-1)((||f||_p)^p + (||g||_p)^p) for two functions in Lp of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p-1) is not needed. Carbery’s question concerns a proposed interpolation between the two situations for p>2. The interpolation parameter measuring the overlap is ||fg||_(p/2). We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all p (joint work with E. A. Carlen, R. L. Frank, and  E. H. Lieb).