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Let $V\equiv (V_1,..., V_M)$ be a random vector with all components continuous random variables. Consider the function $G\colon \mathbb{R}^M \rightarrow \mathbb{R}$ with

$$ G(a)=\mathbb{E}(\max_{k\in \mathcal{M}} (V_k+a_k) )\equiv \int (\max_{k\in \mathcal{M}}( v_k+a_k))f(v)dv $$ for any $a\equiv (a_1,..., a_M)\in \mathbb{R}^M$ and $\mathcal{M}\equiv \{1,...,M\}$, where $\mathbb{E}$ denotes expectation, $v\equiv (v_1,..., v_M)$ is a realisation of $V$, $f$ is the pdf of $V$.

Could you help me to show that $G$ is strictly convex and lower semicontinuous?

Star
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  • Could this help https://math.stackexchange.com/questions/2435464/show-that-max-function-on-rn-is-convex or https://math.stackexchange.com/questions/373229/show-that-the-maximum-of-a-set-of-convex-functions-is-again-convex ? Any further hint? – Star May 24 '18 at 16:17
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    I think the inequality $\max{a + b, c + d} \leq \max{a , c} + \max{b, d}$ for real numbers $a, b, c$ and $d$ is enough to establish the result. – user295959 May 24 '18 at 17:04
  • Could you elaborate more with an answer if you have time? Thank you – Star May 24 '18 at 17:06
  • Because I have the integral which is confusing me. Thank you – Star May 24 '18 at 17:32

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