TITLE: Does star convex imply contractibility?
QUESTION [1 upvotes]: Let $X \subset \mathbb{R}^p$ , $X$ is star convex if there exists a point in $X$ such that for any y in $X$, the line segment joining x and y is contained completely in $X$. 
A set $X \subset \mathbb{R}^p$ is contractible if it can be reduced to one of its points, say P, by a continuous deformation.
Prove if  $ X \subset \mathbb{R}^p $ is star convex then X is contractable.

REPLY [2 votes]: Let $X\subset \mathbb{R}^n$ be a star convex set. Let $x\in X$ be the associated "star point." Consider the function $f:X\times[0,1]\to X$ given by
$$f(y,t) = xt + y(1-t). $$
Then for all $y\in X$, $t\in[0,1]$, $f(y,t)$ lies on the line segment joining $y$ to $x$, and therefore $f(y,t)\in X$. Furthermore, $f(y,0)=y$, and $f(y,1)=x$. Thus $f$ is a deformation of $X$ onto the point $x\in X$. (Oh, I forgot to mention, it is trivial that $f$ is continuous).
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