Minimization problem for convolution 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.)
 A: 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.)
A: (Too long for a comment.)
Numerical experiments suggests that one cannot do better than $1 / \sqrt{\pi x}$ (or one of the equivalent variants, the set of minimizers appears to be quite large). Here is a plot of three minimizers obtained numerically for the discrete analogue of the problem on $\{0, 1, 2, \ldots, n - 1\}$ with $n = 75$. These minimizers were found by Mathematica with three different numerical optimization methods (blue: "DifferentialEvolution", yellow: "NelderMead", green: "SimmulatedAnnealing"). The corresponding norms are 1.12145, 1.12842, 1.1265, respectively.

Mathematica code:
n = 75;
expr = Sum[x[i], {i, 0, n - 1}]/Sqrt[n];
constr = Join[
   Table[Sum[x[j] x[i - j], {j, 0, i}] >= 1, {i, 0, n - 1}], 
   Table[x[i] >= 0, {i, 0, n - 1}]];
vars = Table[x[i], {i, 0, n - 1}];
{min1, subst1} = 
  NMinimize[{expr, constr}, vars, Method -> "DifferentialEvolution"];
{min2, subst2} = 
  NMinimize[{expr, constr}, vars, Method -> "NelderMead"];
{min3, subst3} = 
  NMinimize[{expr, constr}, vars, Method -> "SimulatedAnnealing"];
{min1, min2, min3}
ListPlot[{vars /. subst1, vars /. subst2, vars /. subst3}, 
 Joined -> True, PlotRange -> All]

