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 Fr\'echet mean of radius $R_m>\pi/2$, where $R_m=\pi/2+O(1/m)$. In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius $\pi/2$, however, past the support radius $\pi/2,$ full smeariness may break down, but directional smeariness breaks down only past the support radius $R_m.$ Second, we prove sharpness of this threshold. For every $\varepsilon>0$, we show there exists $m_0(\varepsilon)$ such that for all $m\ge m_0(\varepsilon)$ there exists a rotationally symmetric continuous smeary probability measure on $\mathbb{S}^m$ whose support lies in a ball of radius $\pi/2+\varepsilon$ around the Fr\'echet mean. Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius $\pi/2+\varepsilon$ whose Fr\'echet 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 $\pi/2$ is necessarily Type~I finite sample smeary, i.e. its variance modulation $m_n$ satisfies $\lim_{n\to\infty} m_n>1$. 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.