Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. Suppose that $1\leq p\leq\infty$. If $Y$ is a matrix, then let $\rho(Y)$ denote the spectral radius of $Y$, and let $\|Y\|_{p}$ denote its Schatten $p$-norm whenever $1\leq p\leq\infty$. Suppose that $1\leq p\leq\infty$. Let $L_{\mu,p}:M_n(K)\setminus\{\mathbf{0}\}\rightarrow[-\infty,\infty)$ be the function defined by setting $$L_{\mu,p}(X)=\int\log(\rho(AX))d\mu(A)-\log(\|X\|_p).$$ We observe that $L_{\mu,p}$ is upper semicontinuous and $L_{\mu,p}(\lambda X)=L_{\mu,p}(X)$ for each positive scalar $\lambda$, so by compactness $L_{\mu,p}$ attains its maximum. If $1<p<\infty$, then does there necessarily exist a positive semidefinite $P$ where $\max_{X}L_{\mu,p}(X)=L_{\mu,p}(P)$? What if $\mu$ is supported on the set of all rank $1$ positive semidefinite matrices in $M_n(K)$?
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