Problem 1
open
Let $P_{n}$ be the set of strictly positive definite $n\times n$ real matrices. Define a function $d$ on $P_n \times P_n$ as
$$d(A, B) = \iint_{S^{n-1} \times S^{n-1}} \frac{|u^T(A - B)v|}{(u^TAu)(v^TBv)}dudv,$$
where the integral is taken over all unit norm vectors $u, v \in \mathbb{R}^n$. Prove that
$$d(A + B, C + D) \leq \max(d(A, C), d(B, D)).$$
Source: Cosme Louart.
LaTeX source
Let $P_{n}$ be the set of strictly positive definite $n\times n$ real matrices. Define a function $d$ on $P_n \times P_n$ as
$$d(A, B) = \iint_{S^{n-1} \times S^{n-1}} \frac{|u^T(A - B)v|}{(u^TAu)(v^TBv)}dudv,$$
where the integral is taken over all unit norm vectors $u, v \in \mathbb{R}^n$. Prove that
$$d(A + B, C + D) \leq \max(d(A, C), d(B, D)).$$
Added May 28, 2026.