Is this Proof Correct: If $C$ is a Convex Set with Finite Volume, then $C$ is Bounded?

Question

Question

Prove that if $C$ is a convex set with non-empty interior, and it has finite volume, then $C$ is bounded.

Attempt

Suppose for the sake of contradiction that $C$ is unbounded i.e. for $M>0, \lVert x \rVert >M$ for $x\in M$. Since $C$ is non-empty, there exists at least a point $p\in C$ such that $B(p,\epsilon)\subseteq C$ for $\epsilon >0$. This ball has a positive finite volume.

Since $C$ is convex, consider all points $x^\prime\in C$ such that $x^\prime = (1-\lambda)p+\lambda x = p+\lambda (x-p)$ for $x\in C$. If we scale $C$ from the point $p$ by a factor $\lambda >1$, then $x^\prime$ will still be in $C$. We will have $C^\prime \subseteq C$(since $C$ is unbounded). So, distances within $C$ are scaled by $\lambda$, and the volume of $C$ by $\lambda ^n$, where $n$ is the dimension of the space.

Since by our assumption $C$ is unbounded, we could choose an arbitrarily large $\lambda$ which would scale the volume of any subset with positive volume to an arbitrarily large $\lambda$ which would scale the volume of any subset with positive volume to an arbitrarily large volume. This means the volume of $C$ would be infinite. This contradicts the fact that $C$ has finite volume. So, we reject the assumption that $C$ is unbounded.

Hence, we conclude that $C$ is bounded.

Answer 1

Any convex set will have non-void interior in its affine span. So here we should assume first that the interior of $C$ is nonvoid. Once we have that we can show that if $C$ is unbounded, then $C$ has infinite volume. The reason is that if $C\supset B(0,r)$, and $\|x\|= d$, then the convex hull of $x \cup B(0,r)$ approaches infinity as $d\to \infty$. I think you sketched this argument yourself. I don't have now a clean way to show the fact about the volume, but it it will be larger than the volume of a pyramid with base a circular section of the ball, and height $d$ (that is the intution), so it grows like $\text{const}\times d$.

$\bf{Added:}$ now I see that @Izaak van Dongen aluded to the pyramid too in his comments, so apparently it is a natural thing.