Convex combination and convex hull

Dependencies:

1. Vector Space

Let $V$ be a vector space over $\mathbb{R}$. Let $X = \{x_1, x_2, \ldots, x_n\} \subseteq V$ be a (multi-)set. Then $z \in V$ is called a convex combination of $X$ iff there exist $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that all of the following hold:

• $\lambda_i \in \mathbb{R}$ and $\lambda_i \ge 0$ for all $i$.
• $\sum_{i=1}^n \lambda_i = 1$.
• $z = \sum_{i=1}^n \lambda_ix_i$.

$z \in V$ is called a strict convex combination of $X$ iff all elements of $X$ are distinct and there exist $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that all of the following hold:

• $\lambda_i \in \mathbb{R}$ and $\lambda_i > 0$ for all $i$.
• $\sum_{i=1}^n \lambda_i = 1$.
• $z = \sum_{i=1}^n \lambda_ix_i$.

It is easy to see that if $z$ is a convex combination of $X$, then it is a strict convex combination of a subset of $X$.

The set of all convex combinations of $X$ is called the convex hull of $X$.

Dependency for:

1. Transitivity of convexity
2. Extreme point of a convex set
3. Convex set
4. Extreme direction of convex cone as extreme point of intersection with hyperplane
5. Convex hull of a finite number of points in Euclidean space is bounded

Info:

• Depth: 4
• Number of transitive dependencies: 4

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space