Sharp L1 Inequalities for Sup-Convolution

Let $f$ and $g$ be two real-valued functions defined on a compact convex subset $C$ of $\mathbb R^k$. Their _sup-convolution_ $f*g$ is given by the formula $$f*g(x)=\sup_{(y+z)/2=x}\frac{f(y)+f(z)}2.$$ That is, it is the largest average of $f(y)$ and $f(z)$ over all pairs $y,z$ that average to $x$. The sup-convolution of more functions is defined in a similar way. Sup-convolutions are interesting for many reasons. For instance, if a firm wishes to buy a certain amount $x$ of some resource from $n$ different suppliers and the price from each supplier is a function $p_i$ depends in a non-linear way on the quantity bought (which often happens in practice -- for example, there can be bulk deals), then working out how to minimize the total cost is asking to minimize the quantity $\sum_ip_i(x_i)$ over all $(x_1,\dots,x_n)$ such that $\sum_ix_i=x$. This can easily be transformed into a problem of working out the value of a sup-convolution at $x$. More relevant to this article is a close connection between sup-convolutions and sumsets. The _strict hypograph_ $\text{hyp}(f)$ of a function $f:\mathbb R^k\to\mathbb R$ is the set $\{(x,y:y