Support of Continuous Smeary Measures on Spheres

We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures $\mu$ with density $\rho$ on spheres ${\mathbb{S}}^m$. First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius $R_m>\pi \mathrm{/}2$, where $R_m=\pi \mathrm{/}2+O(1\mathrm{/}m)$. In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius $\pi \mathrm{/}2$, however, past the support radius $\pi \mathrm{/}2,$ full smeariness may break down, but directional smeariness breaks down only past the support radius $R_m\mathrm{.}$ Second, we prove sharpness of this threshold. For every $\epsilon >0$, we show there exists $m_0(\epsilon )$ such that for all $m\ge m_0(\epsilon )$ there exists a rotationally symmetric continuous smeary probability measure on ${\mathbb{S}}^m$ whose support lies in a ball of radius $\pi \mathrm{/}2+\epsilon$ around the Fréchet mean. Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius $\pi \mathrm{/}2+\epsilon$ whose Fréchet function has Hessian of rank one. Finally, we study finite sample smeariness. We show that any continuous non-smeary distribution supported in a geodesic ball of radius $\pi \mathrm{/}2$ is necessarily Type~I finite sample smeary, i.e. its variance modulation $m_n$ satisfies ${\lim }_{n\to \mathrm{\infty }}{m}_n>1$. In the rotationally symmetric case, we further prove a curse-of-dimensionality phenomenon: the variance modulation increases with the dimension and can become arbitrarily large depending on the support.