Suppose $Y$ is a deformation retract of $X$. Then, there exists a homotopy $H:X\times I\rightarrow X$ such that $H(x,0)=x$, $H(x,1)\in Y$, and $H(y,1)=y$ for any $y\in Y$, $x\in X$. So $h(x)=H(x,1)$ is a retraction from $X$ to $Y$. Suppose $i$ is the inclusion from $Y$ to $X$. Then, $h\circ i=Id_Y$. Define $F$ from $X\times I$ to $X$ as $F(x,t)=tx+(1-t)i\circ h(x)$. So, $i\circ h\simeq Id_X$ and $F(y,t)=y$ for any $y\in Y$, which means that Y is a strong deformation retract.

May I ask what is the problem of this proof? If this works, deformation retract will imply strong deformation retract. So there must be something wrong with it.


You're assuming that $X$ is a space where $+$ makes sense, and where every point on the line segment from $x$ to $h(x)$ is contained in $X$.

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    $\begingroup$ So in particular, the proof is perfectly fine for convex subsets of Euclidean space. But... $\endgroup$ – Lee Mosher Sep 24 '19 at 14:00

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