A subspace \(X \subset \mathbb{R}^n\) is said to be star-shaped if there is a point \(x_0 \in X\) such that, for each \(x \in X\), the line segment from \(x_0\) to \(x\) lies in \(X\). Show that if a subspace \(X \subset \mathbb{R}^n\) is locally star-shaped, in the sense that every point of \(X\) has a star-shaped neighborhood in \(X\), then every path in \(X\) is homotopic in \(X\) to a piecewise linear path, that is, a path consisting of a finite number of straight line segments traversed at constant speed. Show this applies in particular when \(X\) is open or when \(X\) is a union of finitely many closed convex sets.
Solution
Let \[ \gamma:I\to X \] be a path.
1. There is a subdivision\[ 0=t_0<t_1<\cdots<t_m=1 \]such that for every \(i\) the set \(\gamma([t_{i-1},t_i])\) lies in a star-shaped neighborhood \(V_i\subseteq X\).
For every \(x\in\gamma(I)\), choose a star-shaped neighborhood \(V_x\) of \(x\) in \(X\). The sets \[ \gamma^{-1}(V_x) \] form an open cover of the compact metric space \(I\). Choose a Lebesgue number \(\delta>0\) for this cover, and choose a subdivision whose mesh is less than \(\delta\). Then each interval \([t_{i-1},t_i]\) is contained in some \(\gamma^{-1}(V_x)\), giving the required \(V_i\).
2. If \(V\subseteq\mathbb R^n\) is star-shaped and \(a,b\in V\), then every path in \(V\) from \(a\) to \(b\) is homotopic relative to its endpoints to a piecewise linear path from \(a\) to \(b\).
Let \(c\in V\) be a star center, so each segment \([c,z]\) with \(z\in V\) lies in \(V\). The map \[ C:V\times I\to V, \qquad C(z,t)=(1-t)z+tc, \] is a contraction of \(V\) to \(c\). Hence every loop in \(V\) is null-homotopic, so \(V\) is simply connected.
Now let \(\alpha\) be any path from \(a\) to \(b\), and let \(p\) be the broken line obtained by traversing the segment from \(a\) to \(c\) and then the segment from \(c\) to \(b\). Both segments lie in \(V\), so \(p\) is a piecewise linear path in \(V\). The loop \(\alpha\cdot\bar p\) is null-homotopic in \(V\). For paths with common endpoints, null-homotopy of \(\alpha\cdot\bar p\) is equivalent to a homotopy \(\alpha\simeq p\) relative to the endpoints: concatenate the null-homotopy with \(p\) and use the standard cancellation homotopies \(\bar p\cdot p\simeq c_b\). Thus \(\alpha\) is homotopic rel endpoints to the piecewise linear path \(p\).
3. The path \(\gamma\) is homotopic relative to its endpoints to a piecewise linear path in \(X\).
For each \(i\), reparametrize the restricted path \[ \gamma|_{[t_{i-1},t_i]} \] to \(I\). By 1 its image lies in the star-shaped set \(V_i\), so 2 gives a homotopy rel endpoints to a broken line \(p_i\subseteq V_i\).
Because each homotopy fixes the two endpoints \(\gamma(t_{i-1})\) and \(\gamma(t_i)\), the finitely many homotopies paste along the subdivision points to a homotopy of \(\gamma\) relative to \(\gamma(0),\gamma(1)\). The terminal path is the concatenation \[ p_1\cdot p_2\cdots p_m, \] which consists of finitely many straight line segments and is therefore piecewise linear.
4. Every open subspace \(X\subseteq\mathbb R^n\) is locally star-shaped.
For \(x\in X\), openness gives \(r>0\) with \[ B(x,r)\subseteq X. \] The ball \(B(x,r)\) is convex, hence star-shaped, and is a neighborhood of \(x\) in \(X\).
5. A finite union of closed convex subsets of \(\mathbb R^n\) is locally star-shaped.
Write \[ X=C_1\cup\cdots\cup C_N \] with each \(C_j\) closed and convex, and fix \(x\in X\). Let \[ J=\{j:x\in C_j\}. \] For every \(j\notin J\), closedness of \(C_j\) and \(x\notin C_j\) give an open ball about \(x\) disjoint from \(C_j\). Since there are only finitely many such \(j\), choose \(r>0\) so that \[ B(x,r)\cap C_j=\varnothing \qquad(j\notin J). \] Then \[ X\cap B(x,r)=\bigcup_{j\in J}(C_j\cap B(x,r)). \] Each \(C_j\cap B(x,r)\) is convex and contains \(x\). Thus if \(y\in X\cap B(x,r)\), then \(y\) belongs to some \(C_j\cap B(x,r)\) with \(j\in J\), and the whole segment \([x,y]\) lies in that same convex set. Therefore \(X\cap B(x,r)\) is star-shaped with center \(x\).
6. Hence the conclusion applies both to open subsets of \(\mathbb R^n\) and to finite unions of closed convex subsets.
Apply 3 using the local star-shapedness established in 4 and 5.