Kinematic formula for Euler characteristicThe relationship between Crofton formula and Radon transform.A min-max formula for depth of the origin in a convex setReference wanted for application of Parametric TransversalityConvergence in the proof of Crofton's FormulaDoes this formula for caliper diameter hold for concave polyhedra?
Kinematic formula for Euler characteristic
The relationship between Crofton formula and Radon transform.A min-max formula for depth of the origin in a convex setReference wanted for application of Parametric TransversalityConvergence in the proof of Crofton's FormulaDoes this formula for caliper diameter hold for concave polyhedra?
$begingroup$
Is there a formula for $int chi(K cap gL) : dg$ (where $chi$ is Euler characteristic) analogous to the kinematic formula for $int mu(K cap gL) : dg$ (where $mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, and $dg$ signifies integration with respect to the Haar measure of that group.
convex-geometry integral-geometry
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
$endgroup$
add a comment
|
$begingroup$
Is there a formula for $int chi(K cap gL) : dg$ (where $chi$ is Euler characteristic) analogous to the kinematic formula for $int mu(K cap gL) : dg$ (where $mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, and $dg$ signifies integration with respect to the Haar measure of that group.
convex-geometry integral-geometry
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
$endgroup$
$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
1
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago
add a comment
|
$begingroup$
Is there a formula for $int chi(K cap gL) : dg$ (where $chi$ is Euler characteristic) analogous to the kinematic formula for $int mu(K cap gL) : dg$ (where $mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, and $dg$ signifies integration with respect to the Haar measure of that group.
convex-geometry integral-geometry
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
$endgroup$
Is there a formula for $int chi(K cap gL) : dg$ (where $chi$ is Euler characteristic) analogous to the kinematic formula for $int mu(K cap gL) : dg$ (where $mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, and $dg$ signifies integration with respect to the Haar measure of that group.
convex-geometry integral-geometry
convex-geometry integral-geometry
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
edited 7 hours ago
Ivan Izmestiev
4,61815 silver badges40 bronze badges
4,61815 silver badges40 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
asked 8 hours ago
James ProppJames Propp
7,2542 gold badges33 silver badges99 bronze badges
7,2542 gold badges33 silver badges99 bronze badges
$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
1
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago
add a comment
|
$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
1
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago
$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
1
1
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago
add a comment
|
1 Answer
1
active
oldest
votes
$begingroup$
Yes, this is called the principal kinematic formula:
$$int chi(K cap gL), dg = sum_k=0^n c_nk V_k(K) V_n-k(L),$$
where $V_i$ are the intrinsic volumes, and $c_nk$ certain constants. See e.g. Section 4.4 in
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.
At the end of that section there are historical references.
Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
$endgroup$
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
add a comment
|
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Yes, this is called the principal kinematic formula:
$$int chi(K cap gL), dg = sum_k=0^n c_nk V_k(K) V_n-k(L),$$
where $V_i$ are the intrinsic volumes, and $c_nk$ certain constants. See e.g. Section 4.4 in
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.
At the end of that section there are historical references.
Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
$endgroup$
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
add a comment
|
$begingroup$
Yes, this is called the principal kinematic formula:
$$int chi(K cap gL), dg = sum_k=0^n c_nk V_k(K) V_n-k(L),$$
where $V_i$ are the intrinsic volumes, and $c_nk$ certain constants. See e.g. Section 4.4 in
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.
At the end of that section there are historical references.
Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
$endgroup$
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
add a comment
|
$begingroup$
Yes, this is called the principal kinematic formula:
$$int chi(K cap gL), dg = sum_k=0^n c_nk V_k(K) V_n-k(L),$$
where $V_i$ are the intrinsic volumes, and $c_nk$ certain constants. See e.g. Section 4.4 in
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.
At the end of that section there are historical references.
Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
$endgroup$
Yes, this is called the principal kinematic formula:
$$int chi(K cap gL), dg = sum_k=0^n c_nk V_k(K) V_n-k(L),$$
where $V_i$ are the intrinsic volumes, and $c_nk$ certain constants. See e.g. Section 4.4 in
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.
At the end of that section there are historical references.
Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
answered 7 hours ago
Ivan IzmestievIvan Izmestiev
4,61815 silver badges40 bronze badges
4,61815 silver badges40 bronze badges
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
add a comment
|
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
$begingroup$
This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research
$endgroup$
– alesia
3 hours ago
add a comment
|
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$begingroup$
I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group.
$endgroup$
– Ben McKay
8 hours ago
1
$begingroup$
$chi(K cap gL) = 1$ if $K cap gL ne emptyset$ and $0$ otherwise.
$endgroup$
– Ivan Izmestiev
7 hours ago