432 \begin{proof} \label{xxyy2} |
432 \begin{proof} \label{xxyy2} |
433 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
433 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
434 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
434 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
435 Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
435 Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
436 Consider the triangle |
436 Consider the triangle |
437 with vertices $z$, $y$ and $s$ with $s\in S$. |
437 with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. |
438 The length of the edge $yz$ is greater than $r$ which is greater |
438 The length of the edge $yz$ is greater than $r$ which is greater |
439 than the length of the edge $ys$. |
439 than the length of the edge $ys$. |
440 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
440 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
441 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
441 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
442 which implies that these points are also in $\Nbd_a(S)$. |
442 which implies that these points are also in $\Nbd_a(S)$. |