Theorem 11.43.6.93F19A4F37D3 Let convex bodies $P_ j,P\subset \Sigma $ contain the same translated simplex $p_*+\delta \Sigma $, with $0<\delta <1$, and suppose that their Hausdorff distance $d_ j=d_ H(P_ j,P)$ tends to zero. Let $r_ j>0$ tend to $r>0$. On the fixed $X$ above, set
\[ (\theta _ j,\varphi _ j)=(r_ j\Theta ,r_ j\pi ^*\Phi _{P_ j}), \qquad (\theta ,\varphi )=(r\Theta ,r\pi ^*\Phi _ P). \]
For every bounded real Borel function $F$, the equation (11.33.1) is solvable in the ordinary type of each $\varphi _ j$ and of $\varphi $; these types have masses $m_ j=r_ j^ n n!\operatorname {Vol}_ n(P_ j)$ and $m=r^ n n!\operatorname {Vol}_ n(P)>0$. Fix a smooth positive volume form $\mathrm{d}V$ on $X$, put $Z_ F=\int _ X\mathrm{e}^ F\, \mathrm{d}V$, and let $B_ j,B$ and $\mathcal A_ j,\mathcal A$ be the corresponding normalization scalars and supporting sets. Then
\[ B_ j(F)=\log m_ j-\log Z_ F,\qquad B(F)=\log m-\log Z_ F. \]
Writing $s=\delta /(n+\sqrt n)$, one has
720678E2362C
11.43.7
\begin{equation} \sup _{G\in C(X,\mathbb R)}|B_ j(G)-B(G)| \leq n|\log (r_ j/r)|+n\log (1+d_ j/s). \end{equation}
Moreover, $n!\operatorname {Vol}_ n(P_ j)\geq \delta ^ n$ and $n!\operatorname {Vol}_ n(P)\geq \delta ^ n$, the backgrounds converge smoothly, and the references converge in $L^1(X)$ and locally uniformly on $\pi ^{-1}(T)$. For continuous real $F$, every supporting set at $F$ equals $\{ \nu _ F\} $, where $\nu _ F=\mathrm{e}^ F\mathrm{d}V/Z_ F$. If continuous $F_ j$ converge uniformly to $F$ and $\nu _ j\in \mathcal A_ j(F_ j)$, then
08069BF2F396
11.43.8
\begin{equation} \int _ X\left|\frac{\mathrm{d}\nu _ j}{\mathrm{d}V} -\frac{\mathrm{d}\nu _ F}{\mathrm{d}V}\right|\mathrm{d}V \leq \mathrm{e}^{2\| F_ j-F\| _\infty }-1. \end{equation}
Proof.
By the two preceding lemmas, each $\varphi _ j$ and $\varphi $ is an envelope fixed point with small unbounded locus and the stated positive mass, and it satisfies the smaller-background condition. Theorem 11.42.1 therefore gives solvability and the formulas for the scalars and supporting sets.
The common simplex contains a ball $c+s\mathbb B$ of radius $s$, the inradius of $\delta \Sigma $, where $\mathbb B$ is the closed unit ball. The Hausdorff distance gives $P_ j\subset P+d_ j\mathbb B$, while $s\mathbb B\subset P-c$ and convexity give
\[ P+d_ j\mathbb B\subset c+(1+d_ j/s)(P-c). \]
Exchanging the roles of $P$ and $P_ j$ bounds both volume ratios by $(1+d_ j/s)^ n$. Subtracting the scalar formulas and including the factor $r_ j^ n/r^ n$ gives (11.43.7). The volume bound holds because $\delta \Sigma $ has volume $\delta ^ n/n!$.
Let $\omega _\ell $ be the modulus of uniform continuity of $\ell $ on $\Sigma $. Matching the maximizers in the definitions of $g_{P_ j}$ and $g_ P$ by points at distance at most $d_ j$ gives
\[ |g_{P_ j}(x)-g_ P(x)|\leq d_ j|x|+\omega _\ell (d_ j). \]
This proves local uniform convergence on $T$. The common lower bound $\chi -\delta \log (n+1)\leq \Phi _{P_ j}$ obtained at the end of the proof of lemma 11.43.2 is integrable on $X$, also after pullback, so boundedness of $r_ j$ and dominated convergence off the coordinate hyperplanes give $L^1$ convergence. Finally, the ratio of the normalized exponential densities lies in $[\mathrm{e}^{-2\eta },\mathrm{e}^{2\eta }]$, where $\eta =\| F_ j-F\| _\infty $. This proves (11.43.8) and hence convergence in total variation.
$\square$
Pome
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