grandes-ecoles 2011 Q12

grandes-ecoles · France · centrale-maths1__pc Proof Proof That a Map Has a Specific Property

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by $$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$ is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by
$$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$
is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

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