Uniqueness of approximate tangent spaces with different multiplicities
Let $M subset mathbb R^{n+k}$ be a $mathcal{H}^n$-measurable subset and $theta : M rightarrow (0,infty)$ be $mathcal{H}^n$ measurable such that for all $K subset mathbb R^{n+k}$ compact we have,
$$ int_{M cap K} theta ,mathrm{d}mathcal{H}^n < infty. $$
We say $M$ has an approximate tangent space $P_x$ at $x in M$ with respect to $theta$ if for all $f in C_c^0(mathbb R^{n+k})$ we have,
$$ int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) rightarrow theta(x) int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) $$
as $lambda rightarrow 0,$ where $eta_{x,lambda}(y) = lambda^{-1}(y-x).$ It is easy to show that $P_x$ is unique if it exists (depending on $M,$ $theta$ and $x$ only), and we usually write $P_x = T_xM.$
Question: Suppose approximate tangent spaces $P_x, tilde P_x$ exist at $x in M$ with respect to two different multiplicity functions $theta$ and $tilde theta.$ Can we conclude that $P_x = tilde P_x?$
This question came to mind when I was reading these lecture notes on GMT by Leon Simon (p41 of PDF); the fact that $theta$ is omitted in the notation of $T_xM$ suggests there is such a uniqueness result.
A partial result: If there exists $C>0$ such that $C^{-1} theta < tilde theta < C theta$ in a neighborhood of $x,$ then the result is true. Indeed if the above holds, then for all $lambda$ sufficiently small we have,
$$ C^{-1}int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < int_{eta_{x,lambda}(M)} f(y) tildetheta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < Cint_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y).$$
Now taking $lambda rightarrow 0$ we get,
$$ C^{-1}theta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) < tildetheta(x)int_{tilde P_x} f(y),mathrm{d}mathcal{H}^n(y) < Ctheta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y). $$
Then it is easy to see that $P_x = tilde P_x,$ by considering appropriate $f$ which are only supported on one of the subspaces.
I'm not sure what happens in the general case however.
Update: It is remarked in Leon Simon's text that we can write almost all of $M$ as a union of increasing compact subsets $M_1 subset M_2 subset dots$ such that each of the restrictions $theta|_{M_j}$ is continuous, and that $T_xM = T_xM_j$ for almost all $x in M_j.$ Combing with above, we see that approximate tangent spaces are unique for almost every point in $M,$ which sufficient for most purposes. I would still be interested in the general case however, particularly if there are counterexamples.
geometric-measure-theory
asked Nov 26 at 13:45
ktoi
2,0981616
add a comment |
Let $M subset mathbb R^{n+k}$ be a $mathcal{H}^n$-measurable subset and $theta : M rightarrow (0,infty)$ be $mathcal{H}^n$ measurable such that for all $K subset mathbb R^{n+k}$ compact we have,
$$ int_{M cap K} theta ,mathrm{d}mathcal{H}^n < infty. $$
We say $M$ has an approximate tangent space $P_x$ at $x in M$ with respect to $theta$ if for all $f in C_c^0(mathbb R^{n+k})$ we have,
$$ int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) rightarrow theta(x) int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) $$
as $lambda rightarrow 0,$ where $eta_{x,lambda}(y) = lambda^{-1}(y-x).$ It is easy to show that $P_x$ is unique if it exists (depending on $M,$ $theta$ and $x$ only), and we usually write $P_x = T_xM.$
Question: Suppose approximate tangent spaces $P_x, tilde P_x$ exist at $x in M$ with respect to two different multiplicity functions $theta$ and $tilde theta.$ Can we conclude that $P_x = tilde P_x?$
This question came to mind when I was reading these lecture notes on GMT by Leon Simon (p41 of PDF); the fact that $theta$ is omitted in the notation of $T_xM$ suggests there is such a uniqueness result.
A partial result: If there exists $C>0$ such that $C^{-1} theta < tilde theta < C theta$ in a neighborhood of $x,$ then the result is true. Indeed if the above holds, then for all $lambda$ sufficiently small we have,
$$ C^{-1}int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < int_{eta_{x,lambda}(M)} f(y) tildetheta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < Cint_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y).$$
Now taking $lambda rightarrow 0$ we get,
$$ C^{-1}theta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) < tildetheta(x)int_{tilde P_x} f(y),mathrm{d}mathcal{H}^n(y) < Ctheta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y). $$
Then it is easy to see that $P_x = tilde P_x,$ by considering appropriate $f$ which are only supported on one of the subspaces.
I'm not sure what happens in the general case however.
Update: It is remarked in Leon Simon's text that we can write almost all of $M$ as a union of increasing compact subsets $M_1 subset M_2 subset dots$ such that each of the restrictions $theta|_{M_j}$ is continuous, and that $T_xM = T_xM_j$ for almost all $x in M_j.$ Combing with above, we see that approximate tangent spaces are unique for almost every point in $M,$ which sufficient for most purposes. I would still be interested in the general case however, particularly if there are counterexamples.
geometric-measure-theory
asked Nov 26 at 13:45
ktoi
2,0981616
add a comment |
Let $M subset mathbb R^{n+k}$ be a $mathcal{H}^n$-measurable subset and $theta : M rightarrow (0,infty)$ be $mathcal{H}^n$ measurable such that for all $K subset mathbb R^{n+k}$ compact we have,
$$ int_{M cap K} theta ,mathrm{d}mathcal{H}^n < infty. $$
We say $M$ has an approximate tangent space $P_x$ at $x in M$ with respect to $theta$ if for all $f in C_c^0(mathbb R^{n+k})$ we have,
$$ int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) rightarrow theta(x) int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) $$
as $lambda rightarrow 0,$ where $eta_{x,lambda}(y) = lambda^{-1}(y-x).$ It is easy to show that $P_x$ is unique if it exists (depending on $M,$ $theta$ and $x$ only), and we usually write $P_x = T_xM.$
Question: Suppose approximate tangent spaces $P_x, tilde P_x$ exist at $x in M$ with respect to two different multiplicity functions $theta$ and $tilde theta.$ Can we conclude that $P_x = tilde P_x?$
This question came to mind when I was reading these lecture notes on GMT by Leon Simon (p41 of PDF); the fact that $theta$ is omitted in the notation of $T_xM$ suggests there is such a uniqueness result.
A partial result: If there exists $C>0$ such that $C^{-1} theta < tilde theta < C theta$ in a neighborhood of $x,$ then the result is true. Indeed if the above holds, then for all $lambda$ sufficiently small we have,
$$ C^{-1}int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < int_{eta_{x,lambda}(M)} f(y) tildetheta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < Cint_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y).$$
Now taking $lambda rightarrow 0$ we get,
$$ C^{-1}theta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) < tildetheta(x)int_{tilde P_x} f(y),mathrm{d}mathcal{H}^n(y) < Ctheta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y). $$
Then it is easy to see that $P_x = tilde P_x,$ by considering appropriate $f$ which are only supported on one of the subspaces.
I'm not sure what happens in the general case however.
Update: It is remarked in Leon Simon's text that we can write almost all of $M$ as a union of increasing compact subsets $M_1 subset M_2 subset dots$ such that each of the restrictions $theta|_{M_j}$ is continuous, and that $T_xM = T_xM_j$ for almost all $x in M_j.$ Combing with above, we see that approximate tangent spaces are unique for almost every point in $M,$ which sufficient for most purposes. I would still be interested in the general case however, particularly if there are counterexamples.
geometric-measure-theory
asked Nov 26 at 13:45
ktoi
2,0981616
Let $M subset mathbb R^{n+k}$ be a $mathcal{H}^n$-measurable subset and $theta : M rightarrow (0,infty)$ be $mathcal{H}^n$ measurable such that for all $K subset mathbb R^{n+k}$ compact we have,
$$ int_{M cap K} theta ,mathrm{d}mathcal{H}^n < infty. $$
We say $M$ has an approximate tangent space $P_x$ at $x in M$ with respect to $theta$ if for all $f in C_c^0(mathbb R^{n+k})$ we have,
$$ int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) rightarrow theta(x) int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) $$
as $lambda rightarrow 0,$ where $eta_{x,lambda}(y) = lambda^{-1}(y-x).$ It is easy to show that $P_x$ is unique if it exists (depending on $M,$ $theta$ and $x$ only), and we usually write $P_x = T_xM.$
Question: Suppose approximate tangent spaces $P_x, tilde P_x$ exist at $x in M$ with respect to two different multiplicity functions $theta$ and $tilde theta.$ Can we conclude that $P_x = tilde P_x?$
This question came to mind when I was reading these lecture notes on GMT by Leon Simon (p41 of PDF); the fact that $theta$ is omitted in the notation of $T_xM$ suggests there is such a uniqueness result.
A partial result: If there exists $C>0$ such that $C^{-1} theta < tilde theta < C theta$ in a neighborhood of $x,$ then the result is true. Indeed if the above holds, then for all $lambda$ sufficiently small we have,
$$ C^{-1}int_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < int_{eta_{x,lambda}(M)} f(y) tildetheta(x+lambda y) ,mathrm{d}mathcal{H}^n(y) < Cint_{eta_{x,lambda}(M)} f(y) theta(x+lambda y) ,mathrm{d}mathcal{H}^n(y).$$
Now taking $lambda rightarrow 0$ we get,
$$ C^{-1}theta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y) < tildetheta(x)int_{tilde P_x} f(y),mathrm{d}mathcal{H}^n(y) < Ctheta(x)int_{P_x} f(y),mathrm{d}mathcal{H}^n(y). $$
Then it is easy to see that $P_x = tilde P_x,$ by considering appropriate $f$ which are only supported on one of the subspaces.
I'm not sure what happens in the general case however.
Update: It is remarked in Leon Simon's text that we can write almost all of $M$ as a union of increasing compact subsets $M_1 subset M_2 subset dots$ such that each of the restrictions $theta|_{M_j}$ is continuous, and that $T_xM = T_xM_j$ for almost all $x in M_j.$ Combing with above, we see that approximate tangent spaces are unique for almost every point in $M,$ which sufficient for most purposes. I would still be interested in the general case however, particularly if there are counterexamples.
geometric-measure-theory
geometric-measure-theory
asked Nov 26 at 13:45
ktoi
2,0981616
asked Nov 26 at 13:45
ktoi
2,0981616
edited Nov 27 at 13:00
asked Nov 26 at 13:45
ktoi
2,0981616
asked Nov 26 at 13:45
ktoi
2,0981616
asked Nov 26 at 13:45
ktoi
2,0981616
2,0981616
add a comment |
add a comment |
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