4 problems

Extremal-value conjecture. The extremal value is

Let $g\in L^1(\mathbb R)\cap L^2(\mathbb R)$ be nonnegative, and let $g\ast g$ denote its autoconvolution. Write $\|\cdot\|_r$ for the $L^r$ norm. The sharp autoconvolution $L^2$ i…

Let $g\in L^1(\mathbb R)\cap L^2(\mathbb R)$ be nonnegative and supported on an interval of length $I$. Write $g\ast g$ for its autoconvolution and $\|\cdot\|_r$ for the $L^r$ norm…

The $\pi/4$ extremal constant conjecture. There is a sequence of Newman polynomials $p$ with $p(1)/\deg(p)\to0$ and $R(p)\to\pi/4$, and $\pi/4$ is the best possible limiting consta…