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I am looking for a simple and short proof showing that $X \to \|X X^\top\|_F^2$ is a convex function where $\|\cdot\|_F$ is the Frobenius norm. I have one proof by showing that the derivative is monotone but it is quite heavy. I expect that there are simple arguments showing this.

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Write $\|XX^\top\|_F^2=g(f(X))$, where $f(X)=\|X\|_4$ is the Schatten 4-norm and $g(t)=t^4$. Since $g$ is increasing on the range of $f$, convexity follows from this answer on MathSE.

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