Continuous boundary of a convex set

Question

Is the boundary of a compact convex set in Rn continuous? Seems like the answer should obviously be yes, but I cannot find any such result in the literature. Can somebody provide a reference (or a proof)? Thanks.

The boundary of any set is a set and as such cannot be continuous. What intuitive property do you have in mind? — Christian Blatter, Jan 27 '13 at 09:36
I'd guess OP intended connected or path-connected rather than continuous? — Benjamin Dickman, Jan 27 '13 at 09:37
Please add this clarification to your question (there is an edit link at the bottom left of your question). You can get a nice $\Bbb R^n$ by writing $\Bbb R^n$ — Martin, Jan 27 '13 at 10:18

score 5 · Answer 1 · answered Jan 27 '13 at 11:05

Let $A\subset \mathbb R^n$ be compact and convex and assume that it is not contained in any $(n-1)$-dimensional affine subspace.

We may assume wlog, that $S^{n-1}\subset A$. The the projection from the origin induces a bijection $S^{n-1}\to \partial A$. One readily checks that this is in fact a homeomorphism and makes $\partial A$ a manifold. Note that for $n=1$ this means that $\partial A$ is not connected.

Continuous boundary of a convex set

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