0
$\begingroup$

I guess that the following statement holds true.

Let $K$ be a convex subset of $\mathbb R^n$ with a positive Lebesgue measure. Then the interior of $K$ is non-empty.

Is a reference or a short proof available?

$\endgroup$
3
$\begingroup$

If the set is a subset of a hyperplane then its Lebesgue measure is $0$, so it is not.

Then $K$ must contain elements $x_0,x_1,\dots,x_n$ that are not in any hyperplane.

Then $K$ contains the open set $\{\sum_{k=0}^n\lambda_kx_k\mid\sum_{k=0}^n\lambda_k=1,\lambda_0,\lambda_1,\dots,\lambda_n>0\}$ as a subset.

$\endgroup$
  • $\begingroup$ Great, that's neat. $\endgroup$ – Bazin Sep 12 '18 at 12:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.