On the existence of minimizer on a log Fano cone singularity

Abstract: We prove that if $x\in (X,Δ,\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and $ν<em>0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A</em>{X,Δ}(ν<em>0)&lt;\infty$, then the value $$δ(X,Δ;ν_0):=\inf</em>{ν\in \mathrm{Val}^{{\mathbb{T},*}_{X,\ni} x}}\frac{A_{X,Δ}(ν)}{S(ν_0;ν)}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.