Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$. Let $V$ be the space of integrable, bounde functions $f:\mathbb{R}\to \mathbb{R}$ that are supported on $[-1/2,1/2]$ and even. We would like to know for which $f\in V$ with $|f|_2=1$ the integral $$I_\epsilon(f) = \int_{-\infty}^\infty \eta(t) (1-f\ast f)(t) dt$$ is minimal, and what that minimum is. Since $\epsilon$ and $\eta$ is fixed, this is the same as the problem of maximizing $\langle \eta,f\ast f\rangle$.

A little calculus of variations shows that, for any extremal $f$ as above, $(\eta \ast f)|_{[-1/2,1/2]}$ has to be a multiple $\lambda f$ of $f$. Moreover, $I_\epsilon(f) = 1 - \langle \eta,f\ast f\rangle = 1 - \langle f\ast \eta,f\rangle = 1-\lambda$. A little work suffices to show that $1-\lambda = O(\epsilon)$.

What is (or are) the extremal $f$, however, and what is $\lambda$? We should at least be able to determine $(1-\lambda)/\epsilon$ as $\epsilon\to 0$.

(Looking at what happens when one takes a high power of the operator $f\mapsto (\eta \ast f)|_{[-1/2,1/2]}$ seems promising, but I suspect there is a concise, standard way of solving this kind of problem.)