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Let $P_1,P_2,\ldots,P_k$ be points in $\mathbb{R}^n$, and let $\mathcal{C}$ denote the convex hull of $P_1,P_2,\ldots,P_k$. Let $P$ be a point inside $\mathcal{C}$. By definition, $P$ can be written as a convex combination of $P_1,P_2,\ldots,P_k$. Caratheodory's Theorem says that we need at most $n+1$ of the points.

What is an example showing that $n+1$ points might be needed? For $n=2$, we can consider a triangle on a plane, and it's clear that to write any interior point as a convex combination we need all three points. But for higher $n$ I'm having trouble visualizing the convex hull, and thus constructing the example.

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In dimension $n$, consider the $n$-dimensional simplex, which is exactly the generalization of a triangle in the plane.

For instance, for $n=3$, take a tetrahedron.

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