PomeLemma 5.29.12.1017EA20E8D8 For every compact $K\subset \mathbb P^ n$,
\[ T(K)\leq \mathrm e\exp \bigl (-\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)^{-1/n}\bigr ). \]
The right-hand side is zero for capacity zero.
Lemma 5.29.121017EA20E8D8
Proof. The pluripolar case follows from the recalled theorem and the characterization of capacity-zero sets. Otherwise let $M=\sup V_ K^*<\infty $. Subtracting $\sup _ Kv$ from a normalized competitor, and conversely normalizing competitors for $V_ K$, shows directly that $T(K)=\mathrm e^{-M}$.
If $M\geq 1$, the function $q=V_ K^*/M$ satisfies $0\leq q\leq 1$ and is an admissible capacity test. Since the background is closed,
\[ 1=\int _ K\omega _{V_ K^*}^ n \leq M^ n\int _ K\omega _ q^ n \leq M^ n\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K). \]
Here $\omega _{V_ K^*}\leq M\omega _ q$ follows from $M\geq 1$. Thus $M\geq \operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)^{-1/n}$. If $M<1$, $V_ K^*$ itself is a capacity test, so the same supported probability gives $\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)=1$. Since $T(K)\leq 1$, the stated factor $\mathrm e$ covers this case. $\square$
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