### Convex set

In an affine space $X$, given a set of points $S\subset X$, the **convex hull** $C(S)$ is defined to be the set consisting of all affine combinations of points $x_1,x_2,\cdots,x_n\in S$ with each $\lambda_i\geq 0$ (and with $\lambda_1+\cdots+\lambda_n=1$, of course). A **convex set** $S\subset X$ is one which is *closed* under affine combinations, such that $C(S)=S$. One could alternatively define "convex set" first as sets which are closed under affine combinations, and then define the "convex hull" $C(S)$ as the smallest convex set containing $S$, i.e. the intersection of all convex sets containing it. (The intersection of arbitrarily many convex sets is convex.)

An affine subspace is always convex, for it is closed under *all* linear combinations in some tangent space , not merely the affine linear combinations.

# geometry

# affine-space

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