Example 18.34.21.A118D36549FE For each $a>0$ and $m\geq 1$, the bound (18.34.20) is optimal as a supremum over smooth weights, already for a selected space of dimension one. Indeed, let $X=\mathbb P^1$ with the trivial line bundle and its trivial metric, take the selecting potential zero, and let $\nu $ be the Fubini–Study area normalized to mass one. Then there are smooth real weights $w_ j$ with $m\operatorname {osc}_ Xw_ j=a$ such that
\[ \| \mu _{w_ j}-\mu _0\| _{\mathrm{TV}}\longrightarrow 2\tanh (a/4). \]
Proof.
The selected space consists of the constant functions, so the normalized kernel probability at a weight $u$ is $\mathrm{e}^{-mu}\nu /\int _ X\mathrm{e}^{-mu}\, \mathrm{d}\nu $. Choose a spherical cap $E$ with $\nu (E)=p=(1+\mathrm{e}^{a/2})^{-1}$; its boundary circle has zero area. Choose smooth functions $0\leq \chi _ j\leq 1$, equal to one on an inner cap and zero outside an outer cap, converging almost everywhere to $\mathbf1_ E$; each has oscillation one. Put $w_ j=-a\chi _ j/m$. The densities $\mathrm{e}^{a\chi _ j}$ converge boundedly almost everywhere, and their normalizing constants converge to a positive limit, so $\mu _{w_ j}$ converges in total variation to
\[ \frac{\mathrm{e}^{a\mathbf1_ E}}{Z}\nu , \qquad Z=1-p+\mathrm{e}^ ap=\mathrm{e}^{a/2}. \]
This limit gives $E$ the mass $1-p$, and its density with respect to $\nu $ is greater than one on $E$ and less than one outside $E$. Its total variation distance from $\nu =\mu _0$ is therefore $2(1-2p)=2\tanh (a/4)$. The triangle inequality transfers this limit to the measures $\mu _{w_ j}$. This shows that the bound is optimal as a supremum; we do not claim that it is attained by a smooth weight.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.