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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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This is plainly wrong as posted here. The filled (d-1)-dimensional unit circle $B_\varepsilon^{d-1}(0)\subset\mathbb{R}^d$ is convex. The smallest affine space (not set, becuase that is not a thing) containing it is $\mathbb{R}^{d-1}$ –  CBenni Jan 29 '13 at 8:36
    
how can be a circle convex??? i guess u r plainly wrong –  Salih Ucan Jan 29 '13 at 8:40
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I di not now what your definition of convex would be, but mine is en.wikipedia.org/wiki/Convex_function. And you will not receive any help from me if you start becoming harsh. On top of that, three question marks are not better than one and correct grammar+spelling help you out, even on the internet. –  CBenni Jan 29 '13 at 8:43
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You're both wrong, methinks. @CBenni said filled circle, also known as a disk. It is convex. On the other hand, I really don't know what a $(d-1)$-dimensional circle is, whether filled or not. But if we assume a $(d-1)$-dimensional ball, it has an empty interior viewed as a subset of $\mathbb{R}^d$, so it is not a counterexample. In fact, the result is true as stated … –  Harald Hanche-Olsen Jan 29 '13 at 8:51
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up vote 2 down vote accepted

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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