grandes-ecoles 2022 Q10

grandes-ecoles · France · x-ens-maths__psi_cpge Proof Proof of Set Membership, Containment, or Structural Property

Let $E$ be a subset of $\mathbb{R}^d$. Recall that $$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$ Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Let $E$ be a subset of $\mathbb{R}^d$. Recall that
$$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$
Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26