Continuity of the Euler characteristic with respect to the Hausdorff metric
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Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance:
$$d_H(X, Y) = max{(
sup_{x in X} inf_{y in Y} d(x,y),
sup_{y in Y} inf_{x in X} d(x,y) )}$$
where $X,Y subseteq mathbb{R}^n$ and $d(cdot,cdot)$ is the Euclidean distance metric.
Now, the zero dimensional intrinsic volume is the Euler characteristic $chi$. I am confused by how $chi$ is continuous with respect to the Hausdorff distance. Consider the following example: two identical balls $A,B$ having diameter $sigma$ are separated by a distance $r$. The Euler characteristic of their union is a valuation, i.e.
$$chi(A cup B) = chi(A) + chi(B) - chi(A cap B)$$
And according to standard texts on integral geometry (e.g. Klain & Rota Introduction to Geometric Probability (2006)) this is continuous with respect to the distance metric above. However, explicitly the Euler characteristic is discontinuous:
$$chi(A cup B) = begin{cases}1 & forall ; r < 2sigma \ 2 & forall ; r > 2sigmaend{cases}$$
whereas the Hausdorff distance between them is simply their separation $d_H(A,B)equiv r$.
How is the Euler characteristic in my counter example continuous with respect to $d_H(A,B)$?
Edit: Mike Miller quite rightly pointed out in the comments that Hadwiger’s theorem applies to strictly convex sets. I am in fact assuming a second extension theorem due to Groemer which generalises the result to so-called polyconvex sets, i.e. sets formed by countable union of convex sets.
geometry differential-geometry algebraic-geometry geometric-probability integral-geometry
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
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|
show 1 more comment
$begingroup$
Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance:
$$d_H(X, Y) = max{(
sup_{x in X} inf_{y in Y} d(x,y),
sup_{y in Y} inf_{x in X} d(x,y) )}$$
where $X,Y subseteq mathbb{R}^n$ and $d(cdot,cdot)$ is the Euclidean distance metric.
Now, the zero dimensional intrinsic volume is the Euler characteristic $chi$. I am confused by how $chi$ is continuous with respect to the Hausdorff distance. Consider the following example: two identical balls $A,B$ having diameter $sigma$ are separated by a distance $r$. The Euler characteristic of their union is a valuation, i.e.
$$chi(A cup B) = chi(A) + chi(B) - chi(A cap B)$$
And according to standard texts on integral geometry (e.g. Klain & Rota Introduction to Geometric Probability (2006)) this is continuous with respect to the distance metric above. However, explicitly the Euler characteristic is discontinuous:
$$chi(A cup B) = begin{cases}1 & forall ; r < 2sigma \ 2 & forall ; r > 2sigmaend{cases}$$
whereas the Hausdorff distance between them is simply their separation $d_H(A,B)equiv r$.
How is the Euler characteristic in my counter example continuous with respect to $d_H(A,B)$?
Edit: Mike Miller quite rightly pointed out in the comments that Hadwiger’s theorem applies to strictly convex sets. I am in fact assuming a second extension theorem due to Groemer which generalises the result to so-called polyconvex sets, i.e. sets formed by countable union of convex sets.
geometry differential-geometry algebraic-geometry geometric-probability integral-geometry
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
$endgroup$
2
$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19
|
show 1 more comment
$begingroup$
Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance:
$$d_H(X, Y) = max{(
sup_{x in X} inf_{y in Y} d(x,y),
sup_{y in Y} inf_{x in X} d(x,y) )}$$
where $X,Y subseteq mathbb{R}^n$ and $d(cdot,cdot)$ is the Euclidean distance metric.
Now, the zero dimensional intrinsic volume is the Euler characteristic $chi$. I am confused by how $chi$ is continuous with respect to the Hausdorff distance. Consider the following example: two identical balls $A,B$ having diameter $sigma$ are separated by a distance $r$. The Euler characteristic of their union is a valuation, i.e.
$$chi(A cup B) = chi(A) + chi(B) - chi(A cap B)$$
And according to standard texts on integral geometry (e.g. Klain & Rota Introduction to Geometric Probability (2006)) this is continuous with respect to the distance metric above. However, explicitly the Euler characteristic is discontinuous:
$$chi(A cup B) = begin{cases}1 & forall ; r < 2sigma \ 2 & forall ; r > 2sigmaend{cases}$$
whereas the Hausdorff distance between them is simply their separation $d_H(A,B)equiv r$.
How is the Euler characteristic in my counter example continuous with respect to $d_H(A,B)$?
Edit: Mike Miller quite rightly pointed out in the comments that Hadwiger’s theorem applies to strictly convex sets. I am in fact assuming a second extension theorem due to Groemer which generalises the result to so-called polyconvex sets, i.e. sets formed by countable union of convex sets.
geometry differential-geometry algebraic-geometry geometric-probability integral-geometry
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
$endgroup$
Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance:
$$d_H(X, Y) = max{(
sup_{x in X} inf_{y in Y} d(x,y),
sup_{y in Y} inf_{x in X} d(x,y) )}$$
where $X,Y subseteq mathbb{R}^n$ and $d(cdot,cdot)$ is the Euclidean distance metric.
Now, the zero dimensional intrinsic volume is the Euler characteristic $chi$. I am confused by how $chi$ is continuous with respect to the Hausdorff distance. Consider the following example: two identical balls $A,B$ having diameter $sigma$ are separated by a distance $r$. The Euler characteristic of their union is a valuation, i.e.
$$chi(A cup B) = chi(A) + chi(B) - chi(A cap B)$$
And according to standard texts on integral geometry (e.g. Klain & Rota Introduction to Geometric Probability (2006)) this is continuous with respect to the distance metric above. However, explicitly the Euler characteristic is discontinuous:
$$chi(A cup B) = begin{cases}1 & forall ; r < 2sigma \ 2 & forall ; r > 2sigmaend{cases}$$
whereas the Hausdorff distance between them is simply their separation $d_H(A,B)equiv r$.
How is the Euler characteristic in my counter example continuous with respect to $d_H(A,B)$?
Edit: Mike Miller quite rightly pointed out in the comments that Hadwiger’s theorem applies to strictly convex sets. I am in fact assuming a second extension theorem due to Groemer which generalises the result to so-called polyconvex sets, i.e. sets formed by countable union of convex sets.
geometry differential-geometry algebraic-geometry geometric-probability integral-geometry
geometry differential-geometry algebraic-geometry geometric-probability integral-geometry
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
edited Dec 8 '18 at 22:37
Joshua Robinson
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
asked Dec 8 '18 at 16:48
Joshua RobinsonJoshua Robinson
312
312
2
$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19
|
show 1 more comment
2
$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19
2
2
$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19
|
show 1 more comment
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$begingroup$
I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof.
$endgroup$
– Mike Miller
Dec 8 '18 at 19:27
$begingroup$
The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:10
$begingroup$
Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here.
$endgroup$
– Mike Miller
Dec 8 '18 at 22:16
$begingroup$
Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question.
$endgroup$
– Joshua Robinson
Dec 8 '18 at 22:31
$begingroup$
Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look.
$endgroup$
– Mike Miller
Dec 9 '18 at 1:19