Generalizing Lee's conjecture on the sum of absolute values of matrices

Abstract: Let $|!\cdot!|p$ denote the Schatten $p$-norm of matrices and $|!\cdot!|_F$ the Frobenius norm. For a square matrix $X$, let $|X|$ denote its absolute value. In 2010, Eun-Young Lee posed the problem of determining the smallest constant $c_p$ such that $|A+B|_p \le c_p|\,|A|+|B|\,|_p$ for all complex matrices $A,B$. The Frobenius case $(p=2)$ conjectured by Lee was proved by Lin and Zhang (2022) and re-proved by Zhang (2025). In this paper, we extend Lee's conjecture to arbitrary numbers of summands and determine the sharp inequality $$ \left|\sum{k=1}^{m} A_k\right|F \le \sqrt{\frac{1+\sqrt{m}}{2}}\; \left|\sum{k=1}^{m}|A_k|\right|_F , $$ with equality attained by an equiangular rank-one family. We further generalize Lee's problem by seeking the smallest constant $c_p(m)$ such that $ |\sum_{k=1}^{m} A_k|p \le c_p(m)\, |\sum{k=1}^{m}|A_k||_p $. It is shown that $c_p(m)\le (\sqrt{m})^{1-1/p}$, and we conjecture a closed-form expression for the optimal value of $c_p(m)$ that recovers all known cases $p=1,2,\infty$.