Franklin's Notes

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.



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