# 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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# 2009-14 New notion on the convexity

Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.

Prove that  a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.

(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)

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