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How to show the following: Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if $\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is the smallest affine space containing convex $C$. Thanks a lot! |
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It can be useful to realize that you can shift $C$ around without changing either its convexity, nor the (non-)emptiness of its interior, nor the property of it affinely spanning $\mathbb{R}$. So we might as well assume $0\in C$, in which case the affine span equals the linear span. If $C$ has a nonempty interior, we can even assume that $0$ is an interior point. in which case the natural basis of $\mathbb{R}^n$, suitably scaled down in size, will lie inside $C$, so $C$ spans all of $\mathbb{R}^n$. On the other hand, assume $C$ spans $\mathbb{R}^n$. Then $C$ contains a basis of $\mathbb{R}^n$. The interior of the simplex with corners in the origin and those basis vectors is contained in $C$, which therefore has nonempty interior. |
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