Elementary Algebraic Topology
Carathéodory's Theorem states that if a point belongs to the convex hull of a set of points in a Euclidean space, then it can be expressed as a convex combination of a certain number of those points. This theorem is crucial in understanding the properties of simplices and simplicial complexes, as it establishes the conditions under which points can be formed from combinations of vertices. It emphasizes the role of dimensionality in geometry and the importance of simplexes in constructing higher-dimensional structures.
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