Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

Abstract: In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $|f+g|p^p \leq 2^{p-1}(|f|_p^p + |g|_p^p)$ for two functions in $L^p$ 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 $p$.