Let a connected reductive group \(G\) act in a vector space \(V\). Suppose \(X\) is a closed \(G\)-stable irreducible subvariety of \(\mathbb P(V)\). Let \(F[X] =\bigoplus_mF[X]_m\) be the homogeneous coordinate ring of \(X\). Consider the decomposition of \(F[X]_m\) as \(G\)-module \(F[X]_m =\bigoplus_{\lambda}\mu_m(\lambda)V^\lambda\), where \(V^\lambda\) is the irreducible \(G\)-module with highest weight \(\lambda\) and \(\mu_m(\lambda)\) are the multiplicities. Let us consider \(\mu_m\) as a measure supported on the weight lattice \(P\) of \(G\). Put \(\varGamma=\text{Convex hull}\left(\bigcup_m(\text{supp}\mu_m)/m\right)\). This is a convex subset of the real vector space \(P\bigotimes_{\mathbb Z}\mathbb R\). It is known that the total mass of \(\mu_m\) is a polynomial in \(m\) for sufficiently large \(m\) (denote by \(k\) its degree) and \(m^{\dim\varGamma -k}\mu_m(m\cdot\lambda)\overset\text{weak}\longrightarrow\mu(\lambda)d\gamma\), where \(d\gamma\) is the Lebesgue measure supported on \(\varGamma\) and the density \(\mu(\lambda)\) is a piecewise-polynomial function.
The aim of this paper is to prove the following: the function \(\mu^{1/(k-\dim\varGamma)}\) is concave on \(\varGamma\); the function \(\log \mu\) is concave on \(P\bigotimes_\mathbb Z\mathbb R\).