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 $H : A \times \left[{0 \,.\,.\, 1}\right] \to A$ 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 $x_0$.


Thanks to the assumption of convexity for $A$, $H$ takes values in $A$; of course $H$ is a continuous function, since it is polynomial separately in $x,t$, and $H(-,0)=\text{id}_A$, $H(-,1)\equiv x_0$ (the constant function on $x_0$). This proves that $H : \text{id}_A \simeq c_{x_0}$. $\blacksquare$