Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\rho(p,q):=\sqrt{\frac12\int_{\mathbb R}(\sqrt{p}-\sqrt q)^2\,dx}=\sqrt{1-H(p,q)}$.
For real $t$, define the $t$-shifted version $p_t$ of $p$ by the formula $p_t(x):=p(x-t)$ for real $x$. A general question is this: Are there broad conditions that guarantee that $H(p_0,p_t)$ will be nonincreasing in $t\ge0$?
A more specific question: if $p$ is unimodal (that is, nondecreasing to the left of some point $c$ and nonincreasing to the right of $c$), will it guarantee that $H(p_0,p_t)$ is nonincreasing in $t\ge0$? This is easy to see if $p$ is also assumed to be symmetric. Numerical experiments (with piecewise-constant $p$) suggest that the unimodality should be enough, even without the symmetry.
(Of course, for (say) saw-like $p$'s, we will not have the desired monotonicity.)
To put this into a context: if one has the strict version of the desired monotonicity, this will allow the convenient reparameterization $[0,\infty)\ni t\mapsto\tau:=\rho(p_0,p_t)$ of the statistical parametric shift (location) family $(p_t)$ of densities.
