372 Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
372 Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
373 \end{lemma} |
373 \end{lemma} |
374 |
374 |
375 \begin{proof} \label{xxyy2} |
375 \begin{proof} \label{xxyy2} |
376 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
376 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
377 Note that $\Nbd_a(S) \sup B_r(y)$. |
377 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
378 Simple applications of the triangle inequality show that $\Nbd_a(S)$ |
378 Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
379 is star-shaped with respect to $y$. |
379 Consider the triangle |
380 \end{proof} |
380 \nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$. |
381 |
381 The length of the edge $yz$ is greater than $r$ which is greater |
382 |
382 than the length of the edge $ys$. |
383 \begin{lemma} \label{xxyy3} |
383 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
384 Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) |
384 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
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385 which implies that these points are also in $\Nbd_a(S)$. |
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386 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
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387 \end{proof} |
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388 |
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389 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
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390 the same result holds, so long as $a$ is not too large: |
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391 |
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392 \begin{lemma} \label{xxzz11} |
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393 Let $M$ be a compact Riemannian manifold. |
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394 Then there is a constant $\rho(M)$ such that for all |
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395 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
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396 $\Nbd_a(S)$ is homeomorphic to a ball. |
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397 \end{lemma} |
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398 |
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399 \begin{proof} |
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400 Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
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401 and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
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402 $y$ distort angles by only a small amount. |
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403 Now the argument of the previous lemma works. |
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404 \end{proof} |
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405 |
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406 |
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407 |
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408 \begin{lemma} \label{xx2phi} |
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409 Let $S \sub M$ be contained in a union (not necessarily disjoint) |
385 of $k$ metric balls of radius $r$. |
410 of $k$ metric balls of radius $r$. |
386 Then there exists a neighborhood $U$ of $S$ such that $U$ is homeomorphic to a disjoint union |
411 Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
387 of balls and |
412 $\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
388 \[ |
413 For convenience, let $\phi_0 = 0$. |
389 \Nbd_{2r}(S) \subeq U \subeq \Nbd_{4^k r}(S) . |
414 Assume also that $\phi_k r \le \rho(M)$. |
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415 Then there exists a neighborhood $U$ of $S$, |
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416 homeomorphic to a disjoint union of balls, such that |
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417 \[ |
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418 \Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
390 \] |
419 \] |
391 \end{lemma} |
420 \end{lemma} |
392 |
421 |
393 \begin{proof} |
422 \begin{proof} |
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423 For $k=1$ this follows from Lemma \ref{xxzz11}. |
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424 Assume inductively that it holds for $k-1$. |
394 Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
425 Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
395 By Lemma \ref{xxyy2}, each $\Nbd(S_i)$ is homeomorphic to a ball. |
426 By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
396 If these balls are disjoint (always the case if $k=1$) we are done. |
427 If these balls are disjoint, let $U$ be their union. |
397 If two (or more) of them intersect, then $S$ is contained in a union of $k-1$ metric |
428 Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
398 balls of radius $4r$. |
429 Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
399 By induction, there is a neighborhood $U$ of $S$ such that |
430 and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
400 \[ |
431 Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
401 U \subeq \Nbd_{4^{k-1}\cdot4r} . |
432 By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
402 \] |
433 homeomorphic to a disjoint union |
403 \end{proof} |
434 of balls, and such that |
404 |
435 \[ |
405 \begin{lemma} \label{xxyy4} |
436 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
406 Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) |
437 \] |
407 of $k$ metric balls of radius $r$. |
438 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
408 Then there exist neighborhoods $U_0, U_1, U_2, \ldots$ of $S$, |
439 \end{proof} |
409 each homeomorphic to a disjoint union of balls, such that |
440 |
410 \[ |
441 |
411 \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) |
442 |
412 \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) |
443 \medskip |
413 \subeq U_2 \subeq \Nbd_{4^{3k} r}(S) \cdots |
444 |
414 \] |
445 |
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446 |
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447 |
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448 |
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449 \noop{ |
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450 |
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451 \begin{lemma} |
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452 |
415 \end{lemma} |
453 \end{lemma} |
416 |
454 |
417 \begin{proof} |
455 \begin{proof} |
418 Apply Lemma \ref {xxyy3} repeatedly. |
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419 \end{proof} |
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420 |
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421 \begin{lemma} \label{xxyy5} |
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422 Let $M$ be a Riemannian $n$-manifold and positive integers $m$ and $k$. |
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423 There exists a constant $\eta(M, m, k)$ such that for all subsets |
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424 $S\subeq M$ which are contained in a (not necessarily disjoint) union of |
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425 $k$ metric balls of radius $r$, $r < \eta(M, m, k)$, |
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426 there exist neighborhoods $U_0, U_1, \ldots, U_m$ of $S$, |
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427 each homeomorphic to a disjoint union of balls, such that |
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428 \[ |
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429 \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) |
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430 \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) \cdots |
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431 \subeq U_m \subeq \Nbd_{4^{(m+1)k} r}(S) . |
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432 \] |
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433 |
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434 \end{lemma} |
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435 |
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436 \begin{proof} |
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437 Choose $\eta = \eta(M, m, k)$ small enough so that metric balls of radius $4^{(m+1)k} \eta$ |
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438 are injective and also have small distortion with respect to a Euclidean metric. |
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439 Then proceed as in the proof of Lemma \ref{xxyy4}. |
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440 \end{proof} |
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441 |
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442 \medskip |
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443 |
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444 |
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445 |
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446 |
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447 |
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448 \noop{ |
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449 |
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450 \begin{lemma} |
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451 |
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452 \end{lemma} |
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453 |
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454 \begin{proof} |
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455 |
456 |
456 \end{proof} |
457 \end{proof} |
457 |
458 |
458 } |
459 } |
459 |
460 |