TITLE: Minimization problem for convolution
QUESTION [11 upvotes]: Let $g(x)$ be a non-negative function supported on $[0,1]$. Let $g \ast g$ denote the convolution of $g$ with itself. Question: What is the smallest possible $L^1(0,1)$ norm of $g$, if I require that $(g \ast g) (t) \geq 1$ for all $t \in [0,1]$. 
Clearly one needs $\|g\|_1 \geq 1$. However, $\|g\|_1=1$ cannot be achieved. But what is the best value that can actually be achieved? 
(Maybe the optimizing function is explicitly known? Something like $g(x) = 1/\sqrt{\pi x}$ works and gives $\|g\|_1 \approx 1.13$, but probably something smaller is possible.)

REPLY [5 votes]: Rick Barnard and I looked at the same problem for the auto-convolution instead of the convolution: arXiv. Our constants are a bit worse because you basically need two square-root singularities, one on each side. These types of inequalities tend to be relevant in combinatorics and they tend to be pretty hard (we discuss some in our paper). One I like a lot can be found in this MO post. (I would have commented but you need 50 reputation for that, sorry.)