Convex Set is Contractible

Theorem
Let $V$ be a topological vector space over $\R$ or $\C$.

Let $A\subset V$ be a convex subset.

Then $A$ is contractible.

Proof
Let $x_0 \in A$.

Define $H : A \times \left[{0 \,.\,.\, 1}\right] \to A$ by:
 * $H \left({x, t}\right) = t x_0 + \left({1 - t}\right) x$

This yields a homotopy between the identity map $\operatorname{id}_A$ and the constant map $x_0$.