grandes-ecoles 2023 Q1

grandes-ecoles · France · x-ens-maths__psi Groups Group Actions and Surjectivity/Injectivity of Maps

Let $C \subset E$ be a convex set. Let $f$ and $g$ be two convex functions from $C$ to $\mathbb{R}$.
(a) Show that $f + g$ is convex, and strictly convex if one of the two functions $f$ or $g$ is strictly convex.
(b) Assume $f$ is strictly convex. Verify that the minimum of $f$ is attained on $C$ at most at one point of $C$.

Let $C \subset E$ be a convex set. Let $f$ and $g$ be two convex functions from $C$ to $\mathbb{R}$.\\
(a) Show that $f + g$ is convex, and strictly convex if one of the two functions $f$ or $g$ is strictly convex.\\
(b) Assume $f$ is strictly convex. Verify that the minimum of $f$ is attained on $C$ at most at one point of $C$.

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