# 2020-16 A convex function of matrices

Let $$A$$ be an $$n \times n$$ Hermitian matrix and $$\lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A)$$ the eigenvalues of $$A$$. Prove that for any $$1 \leq k \leq n$$
$A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)$
is a convex function.

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# 2018-16 A convex function

Find the minimum $$m$$ (if it exists) such that every convex function $$f:[-1,1]\to[-1,1]$$ has a constant $$c$$ such that $\int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.$

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# 2016-6 Convex function

Suppose that $$f$$ is a real-valued convex function on $$\mathbb{R}$$. Prove that the function $$X \mapsto \mathrm{Tr } f(X)$$ on the vector space of $$N \times N$$ Hermitian matrices is convex.

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