I'm reading a publication, where a small bit of convex analysis is used in one of the lemmas. I have Rockafellar's book, but I haven't found anything too helpful yet as I am pretty new to the area.
Lemma. Let $I \subset \{1, \dots, n\}$ for some $n \in \mathbb{N}$ with cardinality $\alpha = \#I$. Define a cone as
$$C = \left\{x \in \mathbb{R}^n : \sum_{i \notin I} |x_i| \leq \sum_{i \in I} |x_i|\right\}.$$ Then for a set
$$ G = \left\{x \in \mathbb{R}^n : \sum_{i=1}^n \mathbb{1}_{\{x_i \neq 0\}} \leq \alpha,~~||x|| = 1\right\},$$ we have that
$$ C \cap \{x \in \mathbb{R}^n : ||x|| = 1\} \subset 3 \cdot \text{Conv}(G),$$ where $||\cdot||$ is the $L^2$ norm.
Proof of Lemma Idea: A Theorem in Rockafellar states that a convex hull is equal to the set of convex combinations of $H$, where for a vector $\lambda \geq 0$ we can define as
$$K(G) := \left\{x \in \mathbb{R}^n : \lambda_1 x_1 + \dots + \lambda_n x_n ~\text{a combination in}~G~\text{s.t.}~~\sum_i^n \lambda_i = 1\right\},$$
and $\text{Conv}(G) = K(G)$.
But I'm not sure how to proceed?