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Prepublicaci´oN´um4,abril2021. DepartamentdeMatem`atiques. http://www.uab.cat/matematiques

INTEGRAL GEOMETRY OF PAIRS OF PLANES

JULI `A CUF´I, EDUARDO GALLEGO AND AGUST´I REVENT ´OS

Abstract. We deal with integrals of invariant measures of pairs of planes in euclidean spaceE3as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke.

1. Introducction

The main goal of this paper is to study integrals of invariant measures with respect to euclidean motions in the euclidean spaceE3, extended to the set of pairs of planes meeting a compact convex set. To carry out this objective we express these integrals in terms of functions of the dihedral visual angle of the convex set from a line and integrate them with respect to an invariant measure in the space of lines.

The first known formula involving the visual angle of a convex set in the eu- clidean planeE2 is Crofton’s formula given in [2]. Other results in this direction were obtained by Hurwitz ([8]), Masotti ([10]) and others, in which the use of Fourier series is the main tool. Recently the authors ([3], [4]) have dealt with a general type of integral formulas from the point of view of Integral Geometry.

When trying to generalize these results to higher dimensions the role played by Fourier series in the case of the plane has to be replaced by the use of spherical harmonics. In this sense Theorem4.1plays an important role. After stating and proving this result in dimension 3 we were aware of the paper by Hug and Schnei- der [7] where a more general result in any dimension is proved. In fact the present paper can be considered in some sense as a complement to [7], the novelty been the introduction of the dihedral visual angle.

In Proposition3.1we give a characterization of invariant measures in the space of pairs of planes. These will be the kind of measures considered along the paper.

In section 4, using Hug-Schneider’s Theorem [7, p. 349], we give an expression for the integral of the sinus of the dihedral visual angle of pairs of planes meet- ing a given compact convex set K in terms of geometrical properties of K, see formula (4.3); also we characterize the compact convex sets of constant width in terms of invariant measures given by Legendre polynomials in Proposition4.3.

In section 5 we assign to any invariant measure on the space of pairs of planes an appropriate function of the dihedral visual angle of a given convex set. The

1991 Mathematics Subject Classification. Primary 52A15, Secondary 53C65.

Key words and phrases. Invariant measures, convex set, visual angle, constant width.

The authors were partially supported by grants 2017SGR358, 2017SGR1725 (Generalitat de Catalunya) and PGC2018-095998-B-100 (FEDER/MICINN).

1

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integral of this function with respect to the measure on the space of lines gives the integral of the above measure extended to those planes meeting the convex set. This result is given in Theorem5.2. Then we relate this result to Blaschke’s work [1]. If K is a convex set of mean curvature M and area of its boundary F , Blaschke proves the formula

Z

G∩K=∅

2− sin2ω) dG = 2M2−π3F 2

where ω = ω(G) is the dihedral visual angle of K from the line G. This equality reveals the significance of the function of the visual angle ω2− sin2ω. One can ask what role does it play the function ω− sin ω; this function, interpreting ω as the visual angle in the plane is significant thanks to Crofton formula. In [1] the

inequality Z

G∩K=∅

(ω− sin ω) dG ≥ π

4(M2− 2πF )

is stablished. Here we provide a simple formulation of the deficit in this inequality by means of Theorem5.4where it is proved that

Z

G∩K=∅

(ω− sin ω)dG = π

4(M2− 2πF ) + π X n=1

Γ(n + 1/2)2

Γ(n + 1)22n(p)k2, whith π2n(p) the projection of the support function p of K on the vector space of spherical harmonics of degree 2n.

In section 6 we give a formulation of Theorem5.2in terms of Fourier series of the function of the visual angle assigned to an invariant measure. As a consequence one obtains that the integral of any invariant measure in the space of pairs of planes extended to those meeting a compact convex set K is an infinite linear combination of integrals of even powers of the sine of the visual angle of K. From this we exhibit in Proposition6.3a simple family of polynomial functions that are in some sense a basis for the integrals considered in Theorem 4.1. In fact every invariant integral can be written as an infinite linear combination of integrals with respect to the invariant measures given by those polynomial functions.

In section 7, motivated by the role played by the integrals of even powers of the sine of the visual angle of a compact set K we compute these integrals in terms of the expansion in spherical harmonics of the support function of K.

2. Preliminaries

Support function. The support function of a compact convex set K in the eu- clidean spaceE3 is defined as

pK(u) = sup{hx, ui : x ∈ K}

for u belonging to the unit sphere S2. If the origin O of E3 is an interior point of K then the number pK(u) is the distance from the origin to the support plane of K in the direction given by u. The width w of K in a direction u ∈ S2 is w(u) = pK(u) + pK(−u).

From now on we will write p(u) = pK(u) and will assume that p(u) is of class C2; in this case we shall say that the boundary of K, ∂K, is of classC2.

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Spherical harmonics. Let us recall that a spherical harmonic of order n on the unit sphere S2 is the restriction to S2 of an harmonic homogeneous polynomial of degree n. It is known that every continuous function on S2 can be uniformly approximated by finite sums of spherical harmonics (see for instance [6]).

More precisely, the function p(u) can be written in terms of spherical harmonics as

p(u) = X n=0

πn(p)(u), (2.1)

where πn(p) is the projection of the support function p on the vector space of spherical harmonics of degree n. An orthogonal basis of this space is given in terms of the longitude θ and the colatitude ϕ in S2 by

{cos(jθ)(sin ϕ)jPn(j)(cos ϕ), sin(jθ)(sin ϕ)jPn(j)(cos ϕ) : 0≤ j ≤ n}

where Pn(j) denotes the jth derivative of the nth Legendre polynomial Pn, see [6].

It can be seen that π0(p) =W/2 = M/4π where W = 1/4πR

S2w(u) du is the mean width of K, and M is the mean curvature of K (cf. [6]). It is clear that π0(p) is invariant under euclidean motions and that π1(p) is not. It is known that πn(p) is also invariant for every n6= 1 (cf. [12]).

As w(u) = p(u) + p(−u), one can easily check that K has constant width if and only if πn(p) = 0 for n6= 0 even.

We recall now the Funk-Hecke theorem which gives the value of the integral over the sphere of a given function multiplied by a spherical harmonic. We restrict ourselves to the case of dimension 3.

Theorem 2.1 (Funk-Hecke). If F : [−1, 1] −→ R is a bounded measurable function and Yn is a spherical harmonic of order n, then

Z

S2

F (hu, vi)Yn(v)dσ(v) = λnYn(u), u∈ S2 with

λn= 2π Z 1

−1

F (t)Pn(t) dt, where Pn is the Legendre polynomial of degree n.

Measures in the space of planes. The space of planes A3,2 inE3 is a homo- geneous space of the group of isometries ofE3. It can also be considered as a line bundle π :A3,2−→Gr(3, 2) where Gr(3, 2) is the Grassmannian of planes through the origin inE3 and π(E) is the plane parallel to E through the origin. The fiber on E0 ∈ Gr(3, 2) is identified with hE0i. Each plane E ∈ A3,2 is then uniquely determined by the pair (π(E), E∩ hπ(E)i). Every pair (E0, p) ∈ Gr(3, 2) × R3 determines an element E0+ p∈ A3,2.

We shall also consider the space of affine lines A3,1 inE3; it is a vector bundle π : A3,1 −→ Gr(3, 1) where Gr(3, 1) is the Grassmannian of lines through the origin and every affine line G⊂ E3can be identified with (π(G), G∩ hπ(G)i) (see for instance [9]).

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Both the isometry group ofE3and the isotropy group of a fixed plane E∈ A3,2

are unimodular groups; so the Haar measure of the group of isometries is projected into a isometry-invariant measure m onA3,2.

For a measurable set B⊂ A3,2 we consider m(B) =

Z

A3,2

χB(E)dE :=

Z

Gr(3,2)

Z

E0

χB(E0+ p)dp

! dν

where χB is the characteristic function of B, dp denotes the ordinary Lebesgue measure on E0 and dν a normalized isometry-invariant measure on Gr(3, 2) such that ν(Gr(3, 2)) = 2π.

More generally, if f : A3,2 → R and ¯f : Gr(3, 2)× R3 → R are related by f (E¯ 0, p) = f (E0+ p) we have

Z

A3,2

f (E) dE :=

Z

Gr(3,2)

Z

E0

f (E¯ 0, p)dp

! dν.

Notice that the only measures onA3,2 invariant under isometries are those of the form f (E)dE with f a constant function.

In a similar way one can define a normalized isometry-invariant measure onA3,1

that will be denoted by dG.

3. Invariant measures in the space of ordered pairs of planes We consider measures in the spaceA3,2×A3,2of pairs of planes inE3of the form mf˜:= ˜f (E1, E2)dE1dE2. We want to study which functions ˜f give an isometry- invariant measure, that is a measure mf˜ satisfying mf˜(B) = mf˜(gB) for every euclidean motion g. For instance, it is known that for a given compact convex set K one hasR

E∩K6=∅dE = M . So when ˜f (E1, E2) = 1 we have Z

K∩Ei6=∅

dE1dE2= M2= 4π2W2, (3.1)

where M andW are the mean curvature and the mean width of K, respectively.

Proposition 3.1. The measure given by ˜f (E1, E2)dE1dE2inA3,2×A3,2is invari- ant under isometries ofE3 if and only if ˜f (E1, E2) = f (hu1, u2i) where π(Ei)= huii, i = 1, 2 and f : [−1, 1] → R is an even measurable function.

Proof. Suppose that ˜f (E1, E2)dE1dE2is invariant. Using the representation of an element E∈ A3,2 as a pair (π(E), p) where p = E∩ hπ(E)i we can write

f (E˜ 1, E2) = F (π(E1), p1; π(E2), p2) for some F : (Gr(3, 2)× E3)2→ R. For any translation τ it is

f (E˜ 1+ τ, E2+ τ ) = F (π(E1), p1+ τ ; π(E2), p2+ τ ).

Due to the invariance of ˜f (E1, E2)dE1dE2 we have

F (π(E1), p1+ τ ; π(E2), p2+ τ ) = F (π(E1), p1; π(E2), p2)

and so F is independent of p1and p2and we can write ˜f (E1, E2) = H(π(E1), π(E2)) for some function H on Gr(3, 2)× Gr(3, 2).

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Given t∈ [−1, 1] consider (V, W ) ∈ Gr(3, 2)2 such that V = hvi, W =hwi and t =hv, wi with v, w unit vectors. The function f(t) = H(V, W ) is well defined since for any rotation θ we have that H(θV, θW ) = H(V, W ) and it is even. So it is proved that there exists a measurable and even function f : [−1, 1] → R such that

f (E˜ 1, E2) = f (hu1, u2i).

If ˜f is as above it is clear that ˜f (E1, E2)dE1dE2gives rise to an isometry-invariant

measure. 

4. Integral of functions of pairs of planes meeting a convex set Let K be a compact convex set in the euclidean space E3. According to equal- ity (3.1) it is a natural question to evaluate

Z

Ei∩K6=∅

f (E˜ 1, E2)dE1 dE2,

where ˜f (E1, E2)dE1dE2 is an isometry-invariant measure on A3,2 × A3,2. This can be done in terms of the coefficients of the expansion of the support function of K in spherical harmonics and the coefficients of the Legendre series of the measurable even function f : [−1, 1] → R such that ˜f (E1, E2) = f (hu1, u2i) (see Proposition3.1).

The following result is a special case, with a different notation, of Theorem 5 in [7]. However we include a proof of this particular case for reader’s convenience.

Theorem 4.1. Let K be a compact convex set with support function p given in terms of spherical harmonics by (2.1). Let ˜f (E1, E2)dE1dE2 be an isometry- invariant measure onA3,2× A3,2 and f : [−1, 1] → R an even measurable function such that ˜f (E1, E2) = f (hu1, u2i) where π(Ei) =huii, i = 1, 2. Then

(4.1)

Z

Ei∩K6=∅

f (E˜ 1, E2)dE1dE2= λ0

4πM2+ X n evenn=2

λnn(p)k2,

where λn = 2πR1

−1f (t)Pn(t) dt with Pn the Legendre polynomial of degree n.

Proof. As ˜f (E1, E2) = f (hu1, u2i) we have that Z

Ei∩K6=∅

f (E˜ 1, E2)dE1 dE2= Z

Ei∩K6=∅

f (hu1, u2i)dE1 dE2. For a fixed plane E2 inA3,2 and writing dEi = dpi∧ dν one has

(4.2) Z

E1∩K6=∅

f (E˜ 1, E2)dE1= Z

Gr(3,2)

Z

hu1i

f (hu1, u2i)dp1

! dν =

= Z

Gr(3,2)

f (hu1, u2i)(p(u1) + p(−u1))dν = Z

S2

f (hu, u2i)p(u) du, where the last equality follows from the fact that f is even.

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As p(u) =P

n=0πn(p)(u) (cf. (2.1)) and using Funk-Hecke’s theorem we have

that Z

E1∩K6=∅

f (E˜ 1, E2)dE1= X n=0

λnπn(p)(u2).

Notice that f being even one has λn= 0 for n odd. Now performing the integral with respect E2 we have

Z

Ei∩K6=∅

f (E˜ 1, E2)dE1 dE2= X n evenn=0

λn Z

Gr(3,2)

Z

hu2i

πn(p)(u2)dp2

! dν =

= X n evenn=0

λn Z

S2

πn(p)(u)p(u) du = X n evenn=0

λnn(p)k2,

where we have used the fact that πn(p)(u) = πn(p)(−u) for n even. Taking into account that π0(p) = M/4π we have

0(p)k2= M2

16π2k1k2= M2 4π and then

Z

Ei∩K6=∅

f (E˜ 1, E2)dE1dE2= λ0 4πM2+

X n evenn=2

λnn(p)k2.

 Example 1. If f (t) = √

1− t2 then f (hu1, u2i) = sin(θ12) where 0 ≤ θ12 ≤ π is the angle between de planes E1 and E2 (that is, cos θ12 = ±hu1, u2i where π(Ei) =huii, i = 1, 2). Applying Theorem4.1with the corresponding coefficients

λ2n = 2π Z 1

−1

f (t)P2n(t) =−Γ(n +12)Γ(n−12) n!(n + 1)!

π

2, λ0= π2, λ2n+1 = 0 (cf. [5], 7.132), one gets

(4.3) Z

Ei∩K6=∅

sin(θ12)dE1dE2= π

4M2−π 2

X n=1

Γ(n +12)Γ(n−12)

n!(n + 1)! kπ2n(p)k2

! . In the particular case that f is a Legendre polynomial one obtains from Theo- rem4.1the following

Corollary 4.2. Let K be a compact convex set with support function p given in terms of spherical harmonics by (2.1). Then if Pn is the Legendre polynomial of even degree n, one has

Z

Ei∩K6=∅

Pn(hu1, u2i) dE1dE2= 4π

2n + 1kπn(p)k2. Proof. In this case λm= 0 for m6= n and λn= 4π

2n + 1. 

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Example 2. As the function f (t) = t2n can be written in terms of Legendre poly- nomials as t2n=

Xn k=0

µn,kP2k(t) with

(4.4) µn,k= (4k + 1)Γ(2n + 1)√π 22n+1Γ(n− k + 1)Γ(n + k + 3/2) (see [5], 8.922) we get the following consequence of Corollary4.2:

Z

Ei∩K6=∅hu1, u2i2ndE1dE2= Xn k=0

4k + 1µn,k2k(p)k2 where µn,k are given by (4.4).

To end this section we analyze equality (4.1) when K is a convex set of constant width. As said this means that πn(p) = 0 for n6= 0 even.

Proposition 4.3. Let K be a compact convex set of constant width W and let f : [−1, 1] −→ R an even bounded measurable function. Then

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2= λ0πW2, (4.5)

with λ0 = 2πR1

−1f (t)dt. Moreover if the above equality holds when f (t) = P2n(t) where P2n is any Legendre polynomial of even degree 2n, n 6= 0 then K is of constant width.

Proof. Since K is of constant width by (4.1) one gets Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2= λ0

4πM2

and remembering that M = 2πW the equality follows. If equality (4.5) holds for f (t) = Pn(t) with n even, n6= 0, and since the corresponding λ0 vanishes one has

Z

Ei∩K6=∅

Pn(hu1, u2i) dE1dE2= 0.

Therefore by Corollary 4.2it follows that kπn(p)k = 0 for every non zero n even

and K is of constant width. 

5. Integrals of invariant measures in terms of the visual angle The aim of this section is to write the integral of a isometry-invariant measure over the pairs of planes meeting a convex set K, given in Theorem 4.1, as an integral of an appropriate function of the visual angle.

Let us precise what we mean by the angle of a plane about a straight line G and the visual angle of a convex set K from a line G not meeting K.

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Definition 5.1.

1. Given a straight line G let (q; e1, e2) be a fixed affine orthonormal frame in G with q ∈ G. For each plane E through G let u be the unit normal vector to E pointing from the origin to it. Then the angle α associated to E is the one given by u = cos(α)e1+ sin(α)e2.

2. The visual angle of a convex set K from a line G not meeting K is the angle ω = ω(G), 0 ≤ ω ≤ π, between the half-planes E1, E2 through G tangents to K.

If αi are the angles associated to Ei, i = 1, 2 then cos(π− ω) = cos(α2− α1) =hu1, u2i

where u1, u2 are the normal unit vectors to E1, E2 pointing from the origin, as- suming the origin inside K.

The measure dE1dE2 in the space A3,2× A3,2 of pairs of planes in E3 can be written according to Santal´o (cf. [11], section II.12.6) as

dE1dE2= sin22− α1) dα12dG.

(5.1)

Then we can prove the following

Theorem 5.2. Let K be a compact convex set and let f : [−1, 1] −→ R be an even continuous function. Let H be the C2 function on [−π, π] satisfying

H00(x) = f (cos(x)) sin2(x), −π ≤ x ≤ π, H(0) = H0(0) = 0.

Then (5.2)

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2= πH(π)F + 2 Z

G∩K=∅

H(ω) dG,

where ui are normal unit vectors to the planes Ei, ω = ω(G) is the visual angle from the line G and F is the area of the boundary of K.

Proof. Let G = q +hui with u a unit director vector such that K ∩ G = ∅.

Let Ei, i = 1, 2 be the supporting planes of K through G. Take now an affine orthonormal frame {q; e1, e2, u} in E3 such that E1 = q +he1, ui. Every plane E through G can be written as E = q +hvα, ui where vα = cos αe1+ sin αe2 with α ∈ [0, π) and the planes E intersecting K correspond to angles α ∈ [0, ω(G)].

Then using (5.1) one has Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2=

= Z

G∩K=∅

Z ω 0

Z ω 0

H002− α1) dα12dG + +

Z

G∩K6=∅

Z π 0

Z π 0

H002− α1) dα12dG.

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Evaluating the inner integrals and taking into account that R

G∩K6=∅dG = π2F it follows

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2=

= 1

2π(H(π) + H(−π))F + Z

G∩K=∅

(H(ω) + H(−ω)) dG.

Since H(0) = H0(0) = 0 and H00(x) = H00(−x) it is easy to see that H(x) = H(−x)

and the result follows. 

Example 3. Consider f (t) = 1 then H(x) = (x2− sin2x)/4 and M2=

Z

Ei∩K6=∅

dE1dE2= 1

3F +1 2

Z

G∩K=∅

2− sin2ω) dG, (5.3)

which gives the well known Blaschke’s formula (cf. [1]) Example 4. Let f (t) =√

1− t2 be the function considered in Example1. In this case the corresponding function H such that

H00(x) = f (cos(x)) sin2(x) =| sin3(x)| is given by

H(x) = 2

3(|x| − | sin x|) −1

9| sin3x|.

Now, since ω∈ [0, π], Theorem 5.2and equality (4.3) leads to Z

G∩K=∅



ω− sin ω − 1 3!sin3ω

 dG =

= π 3M2 16 −1

2πF −3 8

X n=1

Γ(n +12)Γ(n−12)

n!(n + 1)! kπ2n(p)k2

! . Example 5. Taking now f (t) = t2, one gets H(x) = 1/16(x2 − sin2x cos2x) = 1/16(x2−sin2x+sin4x). According Exemple2, Theorem5.2and Blaschke formula one gets

(5.4) 1

3M2+8π

15kπ2(p)k2= Z

Ei∩K6=∅hu1, u2i2dE1dE2=

= 1

4M2+1 8

Z

G∩K=∅

sin4ω dG,

and so Z

G∩K=∅

sin4ω dG = 2

3M2+64π

15 kπ2(p)k2. Notice that in the case that K has constant width one obtains

Z

G∩K=∅

sin4ω dG = 2 3M2. More generally if f (t) = t2n we have the following

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Proposition 5.3. Let K be a compact convex set with mean curvature M. Then one has

(5.5) Z

Ei∩K6=∅hu1, u2i2ndE1dE2=−4A(n)1 M2+ 2

n+1X

r=2

A(n)r Z

G∩K=∅

sin2rω dG, where

A(n)r = (−1)r r−1n 

4r2 3F2(1, r + 1/2, r− n − 1; r, r + 1; 1), r = 1, . . . , n + 1.

We recall that3F2 is the hypergeometric function given by

3F2(a1, a2, a3; b1, b2; z) = X n=0

(a1)n(a2)n(a3)n (b1)n(b2)n

zn n!

with (a)n= Γ(a + n)/Γ(a) is the Pochhammer symbol.

The integral in the left-hand side of (5.5) was calculated in Exemple2in terms of the expansion in spherical harmonics of the support function of K.

Proof. We will apply Theorem 5.2for f (t) = t2n, that is H00(x) = cos2nx sin2x.

We search the corresponding H(x). Le us write H00(x) =

Xn k=0

n k



(−1)ksin2(k+1)x.

Integrating twice and using formula 2.511(2) in [5] we get H(x) = 1

4(x2− sin2x)+

+ Xn k=1

n k

 (−1)k

"

αkx2

2 − 1

2(k + 1)

sin2(k+1)x 2(k + 1) +

Xk r=1

βr(k)sin2(k−r+1)x 2(k− r + 1)

!#

, with

αk= Γ(k + 3/2)

Γ(k + 2) , βr(k)= Γ(k− r + 1)Γ(k + 3/2) Γ(k + 1)Γ(k− r + 3/2). This provides for H(x) an expression of the type

H(x) = A(n)0 x2+ A(n)1 sin2x +· · · + A(n)n+1sin2(n+1)x.

Let us find now the values on A(n)r . Beginning with A(n)0 one gets A(n)0 =

2n n

 22(n+1). For r > 0 one has

A(n)r = 1 4

Xn k=r−1

n k

(−1)k+1 k + 1

βk(k)−r+1

r , r = 1, . . . , n + 1, or in a more compact form

A(n)r = (−1)r r−1n 

4r2 3F2(1, r + 1/2, r− n − 1; r, r + 1; 1), r = 1, . . . , n + 1.

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One easily sees that A(n)0 =−A(n)1 an so (5.6) H(x) = A(n)0 (x2− sin2x) +

n+1X

r=2

A(n)r sin2rx.

Finally Theorem5.2and Blaschke identity (5.3) give the result.  5.1. Crofton’s formula in the space. Blaschke’s formula (5.3) reveals the sig- nificance of the function of the visual angle ω2− sin2ω. One can ask what role the function ω− sin ω plays; this function, interpreting ω as the visual angle in the plane, is significant thanks to Crofton’s formulaR

P /∈K(ω− sin ω)dP = L2/2− πF , where K is a compact convex set in the plane with area F and length of its boundary L (see [11]).

In [1, p. 85] Blaschke shows that (5.7)

Z

G∩K=∅

(ω− sin ω)dG = 1 4

Z

u∈S2

L2udu−π2 2 F, where Lu is the length of the boundary of the projection of K onhui.

It can be easily seen that R

u∈S2Ludu = 2πM and from this equality, applying Schwarz’s inequality, one gets

(5.8)

Z

u∈S2

L2udu≥ πM2. Introducing (5.8) into (5.7) one obtains

(5.9)

Z

G∩K=∅

(ω− sin ω)dG ≥ π

4(M2− 2πF ).

As a consequence of Theorem 5.2we can now evaluate the deficit in both in- equalities (5.8) and (5.9).

Theorem 5.4. Let K be a compact convex set with support function p, area of its boundary F and mean curvature M . Let Lu be the length of the boundary of the projection of K on hui and let ω = ω(G) be the visual angle of K from the line G. Then

i) Z

u∈S2

L2udu = πM2+ 4π X n=1

Γ(n + 1/2)2

Γ(n + 1)22n(p)k2, ii)

Z

G∩K=∅

(ω− sin ω)dG = π

4(M2− 2πF ) + π X n=1

Γ(n + 1/2)2

Γ(n + 1)22n(p)k2. Moreover equality holds both in (5.8) and (5.9) if and only if K is of constant width.

Proof. We consider f (t) = 1/√

1− t2. For this function the corresponding H in Theorem5.2is H(x) =|x|−| sin x|. Applying equality (5.2) and Theorem4.1with the corresponding λ2n’s given by

λ2n= 2π Z 1

−1

f (t)P2n(t)dt = 2πΓ(n + 1/2)2 Γ(n + 1)2 ,

(cf. [5], 7.226), item ii) follows. Equality i) is a consequence of ii) and (5.7).

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The statement about equality in (5.8) and (5.9) is a consequence of the fact that K is of constant width if and only if π2n(p) = 0 for n6= 0. 

6. A formulation with Fourier series

In this section we give an alternative formulation of Theorem 5.2 in terms of Fourier coefficients of the function H00(x). Since f is even one has that H00(x) = f (cos(x)) sin2(x) is an even π-periodic function. Let

(6.1) H00(x) = 1

2a0+X

n≥1

a2ncos(2nx)

be the Fourier expansion of H00(x). Integrating twice and taking into account that H(0) = H0(0) = 0 one obtains

H(x) = a0

4x2+X

n≥1

a2n

4n2(1− cos(2nx)).

(6.2)

Using this expression of the function H, Theorem5.2can be written as

Proposition 6.1. Let K be a compact convex set and let f : [−1, 1] −→ R be an even continuous function. Let H be theC2 function on [−π, π] satisfying

H00(x) = f (cos(x)) sin2(x), −π ≤ x ≤ π, H(0) = H0(0) = 0.

(6.3)

If H(x) is given by (6.2) then (6.4)

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2=

= a0

3F +1 2

Z

G∩K=∅



a0ω2+X

n≥1

a2n

n2 (1− cos(2nω))

 dG,

where ui are normal vectors to the planes Ei, the visual angle from the line G is ω and F denotes the area of the boundary of K.

The right hand side of (6.4) can be written as a linear combination of integrals of even powers of sin ω. For this purpose we will use Blaschke formula (5.3) and the known equality

(6.5) cos 2nx = Xn m=0

αn,msin2mx with αn,m= (−1)mn 22m(n + m− 1)!

(2m)!(n− m)! , which follows easily from the equality cos(2nx) = (−1)nT2n(sin x) where T2n is Chebyshev’s polynomial of degree 2n. We can state

Proposition 6.2. Let K be a compact convex set and let f : [−1, 1] −→ R be an even continuous function. Let H be theC2 function on [−π, π] satisfying

H00(x) = f (cos(x)) sin2(x), −π ≤ x ≤ π, H(0) = H0(0) = 0.

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If H(x) is given by (6.2) then (6.6)

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2=

= a0M2−1 2

X m=2

X n=m

a2n

n2αn,m Z

G∩K=∅

sin2mω dG

! , where ui are normal vectors to the planes Ei, the visual angle from the line G is ω, F denotes the area of the boundary of K, the coefficients αn,m are given by (6.5) and the coefficients a2n by (6.1).

Proof. Using (6.5) the right hand side of (6.4) is written as a0π3

4 F +1 2

Z

G∩K=∅

 a0ω2

X n=1

a2n

n2αn,1sin2ω− X n=2

a2n n2

Xn m=2

αn,msin2mω

 dG =

= a0π3 4 F +1

2 Z

G∩K=∅



a02− sin2ω)− X n=2

a2n

n2 Xn m=2

αn,msin2mω

 dG where we have used that αn,0 = 1, αn,1 = −2n2 and a0 = −2P

n=1a2n which is a consequence of the fact that H00(0) = 0. Using Blaschke formula (5.3) and

reordering the double sum the result follows. 

6.1. A basis for the integrals of invariant measures. As a consequence of Proposition6.2we can exhibit a simple family of polynomial functions that are in some sense a basis for the integrals in Theorem4.1. Consider the polynomials

hm(t) = m(2mt2− 1)(1 − t2)m−2, m > 1.

(6.7)

Then for H00(x) = hm(cos(x)) sin2(x) one easily checks that H(ω) = 12sin2mω and Theorem5.2applied to hm(t) gives

Z

Ei∩K6=∅

hm(hu1, u2i) dE1dE2= Z

G∩K=∅

sin2mω dG, that together with equation (6.6) leads to the following

Proposition 6.3. Under the same hypotheses and notation as in Proposition 6.2 one has

Z

Ei∩K6=∅

f (hu1, u2i) dE1dE2=

= a0M2−1 2

X m=2

X n=m

a2n

n2 αn,m Z

Ei∩K6=∅

hm(hu1, u2i) dE1dE2

! , where the polynomials hmare given in (6.7).

So every invariant integral can be written as an infinite linear combination of the integrals of the invariant measures given by the polynomials hm.

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7. Powers of sine function of the visual angle Equality (6.6) suggests to consider integrals of the form

Z

G∩K6=∅

sin2mω dG.

Integrals of the power of the sine of the visual angle ω for a compact convex set K in the plane were considered in [3] obtaining

Z

P6∈K

sin2mω dP = λ0L2+X

k≥2, even

λkc2k,

for convenient λk and where ck depends on the support function of K. Applying this formula to the projections Kuof a compact convex set K in the euclidean space on the planehui and taking into account that dG = dP ∧ du and ωu(P ) = ω(G) one gets

Z

G∩K=∅

sin2mω dG = 1 2

Z

u∈S2

Z

P6∈Ku

sin2mωudP du =

0

2 Z

u∈S2

L2udu +X

k≥2, even

λk

2 Z

u∈S2

c2k,udu.

The first integral in the right hand side was geometrically interpretated in Theo- rem5.4but the second one has not been handled.

A more direct way to deal with the integral of the power of the sine of the visual angle for a compact convex set K in the euclidean space is by means of our previous results. In fact, we have the following

Proposition 7.1. Let K be a compact convex set with support function p given in terms of spherical harmonics by (2.1) then

(7.1) Z

G∩K=∅

sin2mω dG = m√

π(m− 2)!

4Γ(m +12) M2+

mX−1 k=1

β2k2k(p)k2, m > 1, where M is the mean curvature of K and the β2k’s are given by

(7.2) β2k= (−1)k+1m(m− 2)!2Γ(k + 1/2) (2m− 1)(2k − 1)(k + 1) + m k!(m− k − 1)!Γ(m + k + 1/2) π.

Remark 7.2. When K is a convex set of constant width one haskπ2k(p)k2= 0 and

so Z

G∩K=∅

sin2mω dG = m√π(m− 2)!

4Γ(m + 12) M2. Proof. Comparing relations (4.1) and (5.2) we have

β0

4πM2+ X k=1

β2k2k(p)k2= πH(π)F + 2 Z

G∩K=∅

H(ω) dG where H00(x) = f (cos x) sin2x with H(0) = H0(0) = 0 and β2k= 2πR1

−1f (t)P2k(t) dt.

As said above, the function H corresponding to the polynomials hmgiven in (6.7)

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is H(ω) = 12sin2mω. In this case we have that β2k = 0 for k > m− 1 because hm(t) is a polynomial of degre 2(m− 1) and R1

−1hm(t)P2k(t) = 0 for k > m− 1.

In order to compute

β2k = 2π Z 1

−1

hm(x)P2k(x)dx

we first observe that, using a Computer algebra system (CAS) we get Z 1

−1

hm(x)x2jdx = m(4mj− 2j + 1)Γ(j + 1/2)Γ(m − 1) 2Γ(m + j + 1/2) . (7.3)

Now, recalling the expression of the Legendre polynomials P2k(x) = 1

22k Xk r=0

(−1)r

2k r

4k− 2r 2k

 x2(k−r) we have

Z 1

−1

hm(x)P2k(x)dx =

= mΓ(m− 1) 22k+1

Xk r=0

(−1)r

2k r

4k− 2r 2k

Γ(k−r + 1/2)(4m(k−r)−2(k − r)+1) Γ(m + k− r + 1/2) , but this sum is computable, using again a CAS we have

Z 1

−1

hm(x)P2k(x)dx =

= 1

22k+1mΓ(m− 1)



Γ(m− 1)Γ(1/2 − 2k)Γ(k − 1/2)·

·



(m−1)(2k−1)

4k 2k



((4m−2)k+1)−2(2m−1)k(4k−1)

4k− 2 2k



(2m+2k−1)



·

· 1

2Γ(1/2− k)Γ(m − k)Γ(m + k + 1/2)

 . Simplifyng this expression and taking into account that Γ(1/2 + j)Γ(1/2− j) = (−1)jπ we finally obtain

β2k = 2π Z 1

−1

hm(x)P2k(x)dx

= (−1)k+1m(m− 2)!2Γ(k + 1/2) (2m− 1)(2k − 1)(k + 1) + m k!(m− k − 1)!Γ(m + k + 1/2) π.

 References

[1] W. Blaschke. Vorlesungen uber Integralgeometrie. VEB Deutscher Verlag der Wissenschaften, Berlin, 3rd edition, 1955.

[2] M.W. Crofton. On the theory of local probability. Phil.Trans.R.Soc.Lond., 158:181–199, 1868.

[3] J. Cuf´ı, E. Gallego, and A. Revent´os. Integral geometry about the visual angle of a convex set. Rendiconti del Circolo Matematico di Palermo, 2019.

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[4] J. Cuf´ı, E. Gallego, and A. Revent´os. On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set. Mathematika, 65(4):874–896, 2019.

[5] I.S. Gradshteyn and I.M. Ryzhiz. Table of Integrals, Series and Products. Academic Press, seventh ed edition, 2007.

[6] H. Groemer. Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1996.

[7] D. Hug and R. Schneider. Integral geometry of pairs of hyperplanes or lines. Arch. Math.

(Basel), 115(3):339–351, 2020.

[8] A. Hurwitz. Sur quelques applications geometriques des s´eries de Fourier. Annales scien- tifiques de l’ ´E.N.S., 3`eme s´erie, 19:357–408, dec 1902.

[9] D.A. Klain and G.C. Rota. Introduction to Geometric Probability. Lezioni Lincee. Cambridge University Press, 1997.

[10] G. Masotti. La Geometria Integrale. Rend. Sem. Mat. Fis. Milano, 25:164–231 (1955), 1953–

54.

[11] L.A. Santal´o. Integral geometry and geometric probability. Cambridge Mathematical Library.

Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Mark Kac.

[12] R. Schneider. Convex bodies: the Brunn-Minkowski theory (second expanded edition). Cam- bridge University Press, 2013.

Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia

E-mail address: jcufi@mat.uab.cat, egallego@mat.uab.cat, agusti@mat.uab.cat

Referencias

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