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Let $M_{n}$ denote the set of $n\times n$ real matrices. Let $c>0$ be a real number and denote by $X_1,X_2,\dots,X_n$ the rows of the matrix $X\in M_n$. Let $\|X_i\|$ denote the Euclidean norm of $X_i$.

Let $$M_n^c := \{ X \in M_n :\ \|X_i\|\leq c\ \forall\ i\in\{1,\dots,n\}\}$$ and $\det : M_n \to \mathbb{R}$ be the determinant function. How to maximize $\det$ over $M_n^c$?

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Perform a QR factorization on $X$. Since $Q$ is orthogonal, the columns of $X$ have the same lengths as the columns of $R$. So the length of each column of $R$, and in turn the modulus of each diagonal entry of $R$, is $\le c$. Hence $|\det X|=|\det R|\le c^n$. Edit: The maximum is attained when $X=cQ$ where $Q$ is a special orthogonal matrix (with determinant 1).

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