A compact convex subset is a strong deformation retract

Question

Show that every compact convex subset of $\mathbb{R}^n$ is strong deformation retract.

I don't really know how to approach to this. Any help would be very appreciated. Thanks!

Answer 1

I presume you mean a strong deformation retract of $\Bbb R^n$. Suppose first $A$ is compact convex with an interior point $x_0$. If $v$ is a unit vector, let $\lambda>0$ be the largest number with $x_0+\lambda v\in A$. Define a homotopy on the ray $\{x_0+sv:s\ge0\}$ by $(x_0+sv,t)\mapsto x_0+sv$ when $s\le \lambda$ and $(x_0+sv,t)\mapsto x_0+[(1-t)(s-\lambda)+t\lambda]v$ otherwise. These fit together to give a suitable homotopy; this requires continuity of the Minkowksi functional.

In general $A$ won't have an interior point, but it will lie in an affine subspace in which it has. So first deformation-retract $\Bbb R^n$ to this subspace, then use the above.