Convex Set is Contractible

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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 $H : A \times \closedint 0 1 \to A$ by:

$\map H {x, t} = t x_0 + \paren {1 - t} x$

This yields a homotopy between the identity map $I_A$ and the constant map $x_0$.

Thanks to the assumption of convexity for $A$, $H$ takes values in $A$.

$H$ is a continuous function, since it is polynomial separately in $x, t$, and:

$\map H {-, 0} = I_A$

$\map H {-, 1} \equiv x_0$ (the constant function on $x_0$)

This proves that $H: I_A \simeq c_{x_0}$.

$\blacksquare$