Convex subset with positive measure.

Question

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?

Answer 1

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.