Compact Convex Sets with Nonempty Interior are Homeomorphic

Theorem

Let $n \in \N_{> 0}$.

Let $T, T' \subseteq \R^n$ be compact convex subsets of real Euclidean $n$-space.

Then, $T$ is homeomorphic to $T'$.

Proof

By Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere:

$\partial T \sim \Bbb S^{n - 1}$

$\partial T' \sim \Bbb S^{n - 1}$

Thus, by Homeomorphism Relation is Equivalence:

$\partial T \sim \partial T'$

Hence, by Cones on Homeomorphic Spaces are Homeomorphic:

$C \partial T \sim C \partial T'$

But, by Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary:

$T \sim C \partial T$

$T' \sim C \partial T'$

Therefore, by Homeomorphism Relation is Equivalence:

$T \sim T'$

$\blacksquare$