and the Pompeiu property for polytopes

The null set of a polytope,

and the Pompeiu property for polytopes

Fabr´ıcio C. Machado

joint work with Sinai Robins - arXiv:2104.01957

Instituto de Matem´ atica e Estat´ıstica Universidade de S˜ ao Paulo, Brasil

Harmonic and Spectral Analysis

Acknowledgements

F.C.M was supported by grant #2017/25237-4, from the S˜ ao Paulo

Research Foundation (FAPESP) and was financed in part by the

Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior —

Brasil (CAPES).

Contents

1 Introduction to the Pompeiu problem

2 The Fourier transform of a polytope

3 Bessel functions

4 Overview of the proof

1 Introduction to the Pompeiu problem

Pompeiu property

Let P ⊂ R ^d be a bounded set with nonempty interior. P has the Pompeiu property if, for f ∈ C( R ^d ),

Z

σ(P )

f (x) dx = 0.

over all rigid motions σ ∈ M(d) implies that f ≡ 0.

Pompeiu property

Let P ⊂ R ^d be a bounded set with nonempty interior. P has the Pompeiu property if, for f ∈ C(R ^d ),

Z

σ(P )

f (x) dx = 0.

over all rigid motions σ ∈ M(d) implies that f ≡ 0.

The group M (d) of rigid motions in R ^d is the group generated by

all translations and rotations.

Pompeiu property

Let P ⊂ R ^d be a bounded set with nonempty interior. P has the Pompeiu property if, for f ∈ C( R ^d ),

Z

σ(P )

f (x) dx = 0.

over all rigid motions σ ∈ M(d) implies that f ≡ 0.

Equivalently, P has the Pompeiu property if the values Z

σ(P )

f(x) dx

d

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d = 1

An interval does not have the Pompeiu property:

Z c+L

c

sin 2π

L x

dx = 0

x

d ≥ 2

A ball does not have the Pompeiu property. If R is the radius of the ball and a is such that J _d/2 (aR) = 0, then

Z

kx−ck≤R

sin(ax ₁ ) dx = 0.

R

c

d ≥ 2

A ball does not have the Pompeiu property. If R is the radius of the ball and a is such that J _d/2 (aR) = 0, then

Z

kx−ck≤R

sin(ax 1 ) dx = 0.

Z

kxk≤R

e ^{2πihξ, xi} dx = R kξk

d/2

J _d/2 (2πRkξk).

R

c

Let P ⊂ R ^d be a convex body. P has the Pompeiu property if, for f ∈ C( R ^d ),

Z

σ(P )

f (x) dx = 0.

over all rigid motions σ ∈ M(d) implies that f ≡ 0.

Pompeiu problem

Is the ball the only convex body that does not have the Pompeiu

property?

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

Pompeiu problem

Is the ball the only convex body that does not have the Pompeiu property?

Brown, Schreiber, Taylor (1973): In the planar case, if the

boundary of P has a “corner”, then it has the Pompeiu property.

Williams (1976): If P does *not* have the Pompeiu property

and has a portion of a real analytic surface on its boundary, then

any analytic extension of this surface also lies in the boundary.

Pompeiu problem

Is the ball the only convex body that does not have the Pompeiu property?

Many conditions are known that imply the Pompeiu property:

Brown, Schreiber, Taylor (1973): In the planar case, if the boundary of P has a “corner”, then it has the Pompeiu property.

Williams (1976): If P does *not* have the Pompeiu property

and has a portion of a real analytic surface on its boundary, then

any analytic extension of this surface also lies in the boundary.

Pompeiu problem

Is the ball the only convex body that does not have the Pompeiu property?

Many conditions are known that imply the Pompeiu property:

Brown, Schreiber, Taylor (1973): In the planar case, if the boundary of P has a “corner”, then it has the Pompeiu property.

Williams (1976): If P does *not* have the Pompeiu property and has a portion of a real analytic surface on its boundary, then any analytic extension of this surface also lies in the boundary.

Which in particular implies that polytopes have the Pompeiu property.

Our main contribution

Using an explicit form for the Fourier-Laplace transform of a poly-

tope (Brion’s theorem), we give a simple proof that polytopes have

the Pompeiu property.

Our main contribution

Using an explicit form for the Fourier-Laplace transform of a poly- tope (Brion’s theorem), we give a simple proof that polytopes have the Pompeiu property.

ˆ 1 P (z) = Z

P

e ^−2πihx,zi dx

An equivalent condition

Brown, Schreiber, Taylor (1973)

A convex body P ⊂ R ^d has the Pompeiu property if and only if the

Fourier-Laplace transform of P , namely ˆ 1 _P (z), does not vanish iden-

tically on any of the complex varieties S _C (α), for any α ∈ C \ {0}.

An equivalent condition

Brown, Schreiber, Taylor (1973)

A convex body P ⊂ R ^d has the Pompeiu property if and only if the Fourier-Laplace transform of P , namely ˆ 1 P (z), does not vanish iden- tically on any of the complex varieties S _C (α), for any α ∈ C \ {0}.

S _C (α) = {z ∈ C ^d : z ₁ ² + · · · + z _d ² = α ² }

An equivalent condition

Brown, Schreiber, Taylor (1973)

A convex body P ⊂ R ^d has the Pompeiu property if and only if the Fourier-Laplace transform of P , namely ˆ 1 P (z), does not vanish iden- tically on any of the complex varieties S _C (α), for any α ∈ C \ {0}.

S _C (α) = {z ∈ C ^d : z ₁ ² + · · · + z _d ² = α ² }

Null set of P :

N (P) = {z ∈ C ^d : ˆ 1 _P (z) = 0}

An equivalent condition

Brown, Schreiber, Taylor (1973)

A convex body P ⊂ R ^d has the Pompeiu property if and only if the Fourier-Laplace transform of P , namely ˆ 1 P (z), does not vanish iden- tically on any of the complex varieties S _C (α), for any α ∈ C \ {0}.

S _C (α) = {z ∈ C ^d : z ₁ ² + · · · + z _d ² = α ² }

Null set of P :

N (P) = {z ∈ C ^d : ˆ 1 _P (z) = 0}

Example

Let P ⊂ R ² be an hexagon,

Example

In blue is N (P ) ∩ R ² and in red is a circle.

Example

In blue is N (P ) ∩ R ² and in red is a circle.

Example

In blue is N (P ) ∩ R ² and in red is a circle.

Example

In blue is N (P ) ∩ R ² and in red is a circle.

Our main contribution

Using an explicit form for the Fourier-Laplace transform of a poly- tope (Brion’s theorem), we give a simple proof that polytopes have the Pompeiu property.

Theorem (M., Robins)

Let P ⊂ R ^d be a d-dimensional polytope, H ⊂ R ^d be a 2- dimensional real subspace that is not orthogonal to any edge from P, and fix an orthonormal basis {e, f } ⊂ R ^d for H. Then

α(cos t)e + α(sin t)f ∈ C ^d : t ∈ [−π, π] 6⊂ N (P )

for any α ∈ \ {0}.

2 The Fourier transform of a polytope

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K _v be the tangent cone of P at v and K _v,1 , . . . , K _v,M

be a triangulation of K v into simplicial cones with no new edges. For each 1 ≤ j ≤ M _v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K _v,j .

v

K _v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K v be the tangent cone of P at v and K v,1 , . . . , K v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K v be the tangent cone of P at v and K v,1 , . . . , K v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K v be the tangent cone of P at v and K v,1 , . . . , K v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K v be the tangent cone of P at v and K v,1 , . . . , K v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K _v be the tangent cone of P at v and K _v,1 , . . . , K _v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K _v be the tangent cone of P at v and K _v,1 , . . . , K _v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Let P ⊂ R ^d be a d-dimensional polytope. For each v ∈ V (P ), let K _v be the tangent cone of P at v and K _v,1 , . . . , K _v,M

be a triangulation of K _v into simplicial cones with no new edges. For each 1 ≤ j ≤ M v , let w ^v _j,1 , . . . , w ^v _j,d be the edges of K v,j .

v

Brion’s theorem

ˆ 1 _P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K _v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

Brion’s theorem

ˆ 1 P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

d = 1

P = [− ¹ ₂ , ¹ ₂ ], K ₀ = [− ¹ ₂ , ∞), K ₁ = (−∞, ¹ ₂ ]

−

2 1 2

ˆ 1 _P (z) = e ^−2πi(−

^).z (2πi) ¹

1

z + e ^−2πi

^.z (2πi) ¹

1

(−z) = sin(πz)

πz

Brion’s theorem

ˆ 1 P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

d = 1

P = [− ¹ ₂ , ¹ ₂ ], K ₀ = [− ¹ ₂ , ∞), K ₁ = (−∞, ¹ ₂ ]

−

2 1 2

ˆ 1 _P (z) = e ^−2πi(−

^).z (2πi) ¹

1

z + e ^−2πi

^.z (2πi) ¹

1

(−z) = sin(πz)

πz

Brion’s theorem

ˆ 1 P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

d = 1

P = [− ¹ ₂ , ¹ ₂ ], K ₀ = [− ¹ ₂ , ∞), K ₁ = (−∞, ¹ ₂ ]

−

2 1 2

ˆ 1 _P (z) = e ^−2πi(−

^).z (2πi) ¹

1

z + e ^−2πi

^.z (2πi) ¹

1

(−z) = sin(πz)

πz

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

Brion’s theorem

ˆ 1 P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

d = 2

P = conv{(0, 0), (a, 0), (0, b)},

w ⁰ ₁ = (1, 0), w ^a ₁ = (−1, 0), w ₁ ^b = (0, −1) w ⁰ ₂ = (0, 1), w ^a ₂ = (−a, b), w ₂ ^b = (a, −b)

(0, 0) (a, 0) (0, b)

ˆ 1 _P (z) = 1

2πi 2

1

z z + be ^−2πiaz

(az − bz )z + ae ^−2πibz

(−az + bz )z

Brion’s theorem

ˆ 1 P (z) = X

v∈V (P) M

X

j=1

e ^−2πihv,zi (2πi) ^d

det K v,j

hw ^v _j,1 , zi . . . hw _j,d ^v , zi .

d = 2

P = conv{(0, 0), (a, 0), (0, b)},

w ⁰ ₁ = (1, 0), w ^a ₁ = (−1, 0), w ₁ ^b = (0, −1) w ⁰ ₂ = (0, 1), w ^a ₂ = (−a, b), w ₂ ^b = (a, −b)

(0, 0) (a, 0) (0, b) az

− bz

= 0

ˆ 1 _P (z) = 1

2πi 2

1

z z + be ^−2πiaz

(az − bz )z + ae ^−2πibz

(−az + bz )z

3 Bessel functions

Bessel function of order n, J n (z):

J _n (z) = 1 2π

Z 2π

0

e ^{iz sin t} e ^−int dt

Bessel function of order n, J _n (z):

J n (z) = 1 2π

Z 2π

0

e ^{iz sin t} e ^−int dt

Fourier series expansion for e ^{iz sin t} : e ^{iz sint} = X

n∈ Z

J _n (z)e ^int

Fourier series expansion for e ^{iz sin t} :

e ^{iz sint} = X

n∈ Z

J n (z)e ^int

Fourier series expansion for e ^{iz sin t} :

e ^{iz sint} = X

n∈ Z

J n (z)e ^int

Hypergeometric series J _n (z) = z

2 n ∞

X

k=0

(−1) ^k (n + k)!k!

z 2

2k

Hypergeometric series J n (z) =

z 2

n ∞

X

k=0

(−1) ^k (n + k)!k!

z 2

2k

Hypergeometric series

J _n (z) = z 2

n ∞

X

k=0

(−1) ^k (n + k)!k!

z 2

2k

n→∞ lim J n (z) 1

n!

z 2

n −1

= 1

Hypergeometric series

J _n (z) = z 2

n ∞

X

k=0

(−1) ^k (n + k)!k!

z 2

2k

n→∞ lim J n (z) 1

n!

z 2

n −1

= 1

4 Overview of the proof

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

Brown, Schreiber, Taylor (1973)

P does not have Pompeiu property ⇐⇒ ∃α 6= 0 : S _C (α) ⊂ N (P )

z(t) = (z 1 , . . . , z d ) ∈ C ,

z 1 = α cos t, z 2 = α sin t, z 3 = · · · = z _d = 0, t ∈ (−π, π]

Assume by contradiction that

ˆ 1 _P (z(t)) = 0 for all t ∈ (−π, π].

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

Brown, Schreiber, Taylor (1973)

P does not have Pompeiu property ⇐⇒ ∃α 6= 0 : S _C (α) ⊂ N (P )

’circle’

z(t) = (z 1 , . . . , z d ) ∈ C ^d ,

z ₁ = α cos t, z ₂ = α sin t, z ₃ = · · · = z _d = 0, t ∈ (−π, π]

1 _P (z(t)) = 0 for all t ∈ (−π, π].

Brown, Schreiber, Taylor (1973)

P does not have Pompeiu property ⇐⇒ ∃α 6= 0 : S _C (α) ⊂ N (P )

’circle’

z(t) = (z 1 , . . . , z d ) ∈ C ^d ,

z ₁ = α cos t, z ₂ = α sin t, z ₃ = · · · = z _d = 0, t ∈ (−π, π]

Assume by contradiction that

ˆ 1 _P (z(t)) = 0 for all t ∈ (−π, π].

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = ˆ 1 P (z(t)) = X

v∈V (P) M

X

j=1

e −2πihv, z(t)i

(2πi) ^d

det K _v,j

hw _j,1 ^v , z(t)i . . . hw _j,d ^v , z(t)i

j,l

0 = P

v∈V (P )

P N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

c _u,N 6= 0

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = ˆ 1 P (z(t)) = X

v∈V (P) M

X

j=1

e −2πihv, z(t)i

(2πi) ^d

det K _v,j

hw _j,1 ^v , z(t)i . . . hw _j,d ^v , z(t)i

Substituting cos t = (e ^it + e ^−it )/2 and sin t = (e ^it − e ^−it )/(2i), hw _j,l ^v , z(t)i is a trigonometric polynomial c −1 e ^−it + c 0 + c 1 e ^it

c _u,N 6= 0

0 = ˆ 1 P (z(t)) = X

v∈V (P) M

X

j=1

e −2πihv, z(t)i

(2πi) ^d

det K _v,j

hw _j,1 ^v , z(t)i . . . hw _j,d ^v , z(t)i

Substituting cos t = (e ^it + e ^−it )/2 and sin t = (e ^it − e ^−it )/(2i), hw _j,l ^v , z(t)i is a trigonometric polynomial c −1 e ^−it + c 0 + c 1 e ^it

0 = P

v∈V (P )

P N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

c _u,N 6= 0

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r _v cos φ _v , r _v sin φ _v , v ₃ , . . . , v _d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i =

e ^−2πiαr

^cos(t−φ

⁾ =

e ^2πiαr

^sin(t−φ

^−π/2)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i =

e ^−2πiαr

^cos(t−φ

⁾ =

e ^2πiαr

^sin(t−φ

^−π/2)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i

=

e ^−2πiαr

^cos(t−φ

⁾ =

e ^2πiαr

^sin(t−φ

^−π/2)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i

= e ^−2πiαr

^cos(t−φ

⁾ =

e ^2πiαr

^sin(t−φ

^−π/2)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i

= e ^−2πiαr

^cos(t−φ

⁾ = e ^2πiαr

^sin(t−φ

^−π/2)

= X

n∈ Z

J n (2παr v )e ^int e ^−in(φ

^+π/2)

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i

= e ^−2πiαr

^cos(t−φ

⁾ = e ^2πiαr

^sin(t−φ

^−π/2)

= X

J n (2παr v )e ^int e ^−in(φ

^+π/2)

0 = P

v∈V (P )

P _N

k=−N c _v,k e ^ikt

e −2πihv, z(t)i

e ^{iz sint} = P

n∈ Z J n (z)e ^int

v = (r v cos φ v , r v sin φ v , v 3 , . . . , v d ) z(t) = (α cos t, α sin t, 0, . . . , 0)

e −2πihv, z(t)i

= e ^−2πiαr

^cos(t−φ

⁾ = e ^2πiαr

^sin(t−φ

^−π/2)

= X

J n (2παr v )e ^int e ^−in(φ

^+π/2)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = X

v∈V (P )

X ^N

k=−N

c v,k e ^ikt

e −2πihv, z(t)i

= X

v∈V (P )

X ^N

k=−N

c _v,k e ^ikt X

n∈ Z

J _n (2παr _v )e ^int e ^−in(φ

^+π/2)

n∈ Z v∈V (P ) k=−N

0 = X

v∈V (P )

e ^−inφ

N

X

k=−N

c _v,k J n−k (2παr _v )i ^k e ^ikφ

, ∀n ∈ Z

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

0 = X

v∈V (P )

X ^N

k=−N

c v,k e ^ikt

e −2πihv, z(t)i

= X

v∈V (P )

X ^N

k=−N

c _v,k e ^ikt X

n∈ Z

J _n (2παr _v )e ^int e ^−in(φ

^+π/2)

= X

n∈ Z

X

v∈V (P ) N

X

k=−N

c v,k e ^{−i(n−k)(φ}

^+π/2) J n−k (2παr v )

e ^int

v∈V (P ) k=−N

0 = X

v∈V (P )

X ^N

k=−N

c v,k e ^ikt

e −2πihv, z(t)i

= X

v∈V (P )

X ^N

k=−N

c _v,k e ^ikt X

n∈ Z

J _n (2παr _v )e ^int e ^−in(φ

^+π/2)

= X

n∈ Z

X

v∈V (P ) N

X

k=−N

c v,k e ^{−i(n−k)(φ}

^+π/2) J n−k (2παr v )

e ^int

0 = X

e ^−inφ

N

X

k=−N

c _v,k J _n−k (2παr _v )i ^k e ^ikφ

, ∀n ∈ Z

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

By translating P in the direction of u, we may assume that u = arg max

v∈V r _v

and that u is the only vertex that attains this maximum.

n→∞ (2παr u ) 0 if k < N or (k = N and u 6= v)

By translating P in the direction of u, we may assume that u = arg max

v∈V r _v

and that u is the only vertex that attains this maximum.

n→∞ lim

(n − N )!2 ^n−N

(2παr u ) ^n−N J n−k (2παr v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

By translating P in the direction of u, we may assume that u = arg max

v∈V r _v

and that u is the only vertex that attains this maximum.

n→∞ lim

(n − N )!2 ^n−N

(2παr u ) ^n−N J n−k (2παr v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

n→∞ lim

(n − N )!2 ^n−N

(2παr _u ) ^n−N J n−k (2παr _v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

n→∞ lim

(n − N )!2 ^n−N

(2παr _u ) ^n−N J n−k (2παr v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

0 = X

v∈V (P )

e ^−inφ

N

X

k=−N

c _v,k J n−k (2παr v )i ^k e ^ikφ

n→∞ lim

(n − N )!2 ^n−N

(2παr _u ) ^n−N J n−k (2παr v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

0 = X

v∈V (P )

e ^−inφ

N

X

k=−N

c _v,k J n−k (2παr v )i ^k e ^ikφ

n→∞ lim ×e ^inφ

^(n−N)!2 _(2παr

u

)

Introduction to the Pompeiu problem The Fourier transform of a polytope Bessel functions Overview of the proof

n→∞ lim

(n − N )!2 ^n−N

(2παr _u ) ^n−N J n−k (2παr _v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

0 = X

v∈V (P )

e ^−inφ

N

X

k=−N

c _v,k J n−k (2παr _v )i ^k e ^ikφ

n→∞ lim ×e ^inφ

^(n−N)!2

(2παr

)

= ⇒ c u,N i ^N e ^{iN φ}

= 0 = ⇒ c u,N = 0

n→∞ lim

(n − N )!2 ^n−N

(2παr _u ) ^n−N J n−k (2παr _v ) =

( 1 if k = N and u = v,

0 if k < N or (k = N and u 6= v)

0 = X

v∈V (P )

e ^−inφ

N

X

k=−N

c _v,k J n−k (2παr _v )i ^k e ^ikφ

n→∞ lim ×e ^inφ

^(n−N)!2

(2παr

)

= ⇒ c u,N i ^N e ^{iN φ}

= 0 = ⇒ c u,N = 0 a contradiction.

Summary

Brion’s theorem shows the Fourier-Laplace transform of a polytope as a rational-exponential function.

The null set of a polytope P does not contain a ’circle’ on any plane not orthogonal to some edge of P.

To show this we assume otherwise and find a contradiction between the Fourier coefficients of ˆ 1 _P (z(t)) of high order and the asymptotic behaviour of Bessel functions.

This implies that every polytope has the Pompeiu property.

Selected References

[1] R. Benguria, M. Levitin, and L. Parnovski.

Fourier transform, null variety, and Laplacian’s eigenvalues.

J. Funct. Anal., 257(7):2088–2123, 2009.

[2] L. Brown, B. M. Schreiber, and B. A. Taylor.

Spectral synthesis and the Pompeiu problem.

Ann. Inst. Fourier (Grenoble), 23(3):125–154, 1973.

[3] A. G. Ramm.

Symmetry problems. The Navier-Stokes problem.

Synthesis lectures on Mathematics and Statistics. Morgan & Claypool, 2019.

[4] S. A. Williams.

A partial solution of the Pompeiu problem.

Math. Ann., 223(2):183–190, 1976.

Thank you for your attention!

Fabr´ ıcio Caluza Machado www.ime.usp.br/∼fabcm [email protected]

Mathematics and Statistics Institute,

University of S~ ao Paulo, Brazil