Definition of Convex Hull
Definition 1
The convex hull $C$ of a subset $X$ of a vector space $V$ refers to the intersection of all convex sets that contain $X$, and is mathematically defined as follows. $$ C = \left\{ \sum_{k} t_{k} \mathbf{x}_{k} : \mathbf{x}_{k} \in X, t_{k} \ge 0 , \sum_{k} t_{k} = 1 \right\} $$
Explanation
The equation mentioned in the definition isn’t exactly a definition per se. The expression $$ \sum_{k} t_{k} \mathbf{x}_{k} $$ written in set-builder notation within the set symbols, is called the convex combination of $\left\{ \mathbf{x}_{k} \right\}_{k}$.
Though it may sound complicated, it’s quite simple when seen through diagrams2, and the term convex hull is not so much important in itself but rather suddenly appears in contexts where one wishes to easily handle spaces in geometry, optimization theory, topological data analysis, etc.
The convex hull illustrated is really simple. It is the smallest convex set that encloses all points of $X$. The reason why it talks about the intersection of all convex sets in a mathematical sense is because expressions like ‘size’ being ‘small’ are not really intuitive in mathematics.
Matousek. (2007). Understanding and Using Linear Programming: p49. ↩︎
Sheffar. (2020). Introductory Topological Data Analysis. https://arxiv.org/abs/2004.04108v1 ↩︎