All quasi-isometry is coarsely surjective.
$begingroup$
Definition. $(X,d_X)$, $(Y,d_Y)$ metric spaces.
$f:Xto Y$ is quasi-isometry if
$d_Y(f(x),f(x'))leq Ld_X(x,x')+C$
exists coarse inverse, i.e. $overline{f}:Yto X$ such that
$d_X(overline{f}f(x),x)leq C$
$d_Y(foverline{f}(y),y)leq C$
and $d_X(overline{f}(y),overline{f}(y'))leq Ld_Y(y,y')+C$
for all $x,x'in X$, for all $y,y'in Y$
How prove that $f:Xto Y$ ($L-C$) quasi-isometry then $f$ is coarsely surjective? i.e. $forall yin Y, exists f(x)in f(X)$ such that $d_Y(y,f(x))leq C$
metric-spaces lipschitz-functions
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
$endgroup$
add a comment |
$begingroup$
Definition. $(X,d_X)$, $(Y,d_Y)$ metric spaces.
$f:Xto Y$ is quasi-isometry if
$d_Y(f(x),f(x'))leq Ld_X(x,x')+C$
exists coarse inverse, i.e. $overline{f}:Yto X$ such that
$d_X(overline{f}f(x),x)leq C$
$d_Y(foverline{f}(y),y)leq C$
and $d_X(overline{f}(y),overline{f}(y'))leq Ld_Y(y,y')+C$
for all $x,x'in X$, for all $y,y'in Y$
How prove that $f:Xto Y$ ($L-C$) quasi-isometry then $f$ is coarsely surjective? i.e. $forall yin Y, exists f(x)in f(X)$ such that $d_Y(y,f(x))leq C$
metric-spaces lipschitz-functions
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
$endgroup$
add a comment |
$begingroup$
Definition. $(X,d_X)$, $(Y,d_Y)$ metric spaces.
$f:Xto Y$ is quasi-isometry if
$d_Y(f(x),f(x'))leq Ld_X(x,x')+C$
exists coarse inverse, i.e. $overline{f}:Yto X$ such that
$d_X(overline{f}f(x),x)leq C$
$d_Y(foverline{f}(y),y)leq C$
and $d_X(overline{f}(y),overline{f}(y'))leq Ld_Y(y,y')+C$
for all $x,x'in X$, for all $y,y'in Y$
How prove that $f:Xto Y$ ($L-C$) quasi-isometry then $f$ is coarsely surjective? i.e. $forall yin Y, exists f(x)in f(X)$ such that $d_Y(y,f(x))leq C$
metric-spaces lipschitz-functions
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
$endgroup$
Definition. $(X,d_X)$, $(Y,d_Y)$ metric spaces.
$f:Xto Y$ is quasi-isometry if
$d_Y(f(x),f(x'))leq Ld_X(x,x')+C$
exists coarse inverse, i.e. $overline{f}:Yto X$ such that
$d_X(overline{f}f(x),x)leq C$
$d_Y(foverline{f}(y),y)leq C$
and $d_X(overline{f}(y),overline{f}(y'))leq Ld_Y(y,y')+C$
for all $x,x'in X$, for all $y,y'in Y$
How prove that $f:Xto Y$ ($L-C$) quasi-isometry then $f$ is coarsely surjective? i.e. $forall yin Y, exists f(x)in f(X)$ such that $d_Y(y,f(x))leq C$
metric-spaces lipschitz-functions
metric-spaces lipschitz-functions
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
edited Nov 7 '18 at 15:58
eraldcoil
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
asked Nov 7 '18 at 15:53
eraldcoileraldcoil
388211
388211
add a comment |
add a comment |
1 Answer
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$begingroup$
Put $x=bar f(y)$.
..........
Given $yin Y$ we need to find $x$ satisfying given property. But the definition of a quasi-isometry trivially implies that $x=bar f(y)$ fits.
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Put $x=bar f(y)$.
..........
Given $yin Y$ we need to find $x$ satisfying given property. But the definition of a quasi-isometry trivially implies that $x=bar f(y)$ fits.
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
$endgroup$
add a comment |
$begingroup$
Put $x=bar f(y)$.
..........
Given $yin Y$ we need to find $x$ satisfying given property. But the definition of a quasi-isometry trivially implies that $x=bar f(y)$ fits.
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
$endgroup$
add a comment |
$begingroup$
Put $x=bar f(y)$.
..........
Given $yin Y$ we need to find $x$ satisfying given property. But the definition of a quasi-isometry trivially implies that $x=bar f(y)$ fits.
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
$endgroup$
Put $x=bar f(y)$.
..........
Given $yin Y$ we need to find $x$ satisfying given property. But the definition of a quasi-isometry trivially implies that $x=bar f(y)$ fits.
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
edited Jan 3 at 17:49
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
answered Dec 5 '18 at 8:11
Alex RavskyAlex Ravsky
39.6k32181
39.6k32181
add a comment |
add a comment |
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