A note on the area and coarea formula

A note on the area and coarea formula

Daniel Cibotaru joint with Jorge de Lira

Universidade Federal do Ceará

7th International Meeting on Lorentzian Geometry USP, July 2013

(UFC) 1 / 20

An "easy" question

Letπ :P →M be a fiber bundle with smooth fiber and compact baseM.

Question: When does

vol(s(M))≥vol(M) hold for every sections:M→P?

Possible answer: WhenP→Bis a Riemannian submersion.

vols(M) = Z

M

√

detds^∗ds dvol_M ≥ Z

M

dvol_M ds=id⊕d˜s⇒ds^∗ds=id+d˜s^∗ds˜≥id.

(UFC) 2 / 20

An "easy" question

Letπ :P →M be a fiber bundle with smooth fiber and compact baseM.

Question: When does

vol(s(M))≥vol(M) hold for every sections:M→P?

Possible answer: WhenP→Bis a Riemannian submersion.

vols(M) = Z

M

√

detds^∗ds dvol_M ≥ Z

M

dvol_M ds=id⊕d˜s⇒ds^∗ds=id+d˜s^∗ds˜≥id.

(UFC) 2 / 20

An "easy" question

Letπ :P →M be a fiber bundle with smooth fiber and compact baseM.

Question: When does

vol(s(M))≥vol(M) hold for every sections:M→P?

Possible answer: WhenP→Bis a Riemannian submersion.

vols(M) = Z

M

√

detds^∗ds dvol_M ≥ Z

M

dvol_M ds=id⊕d˜s⇒ds^∗ds=id+d˜s^∗ds˜≥id.

(UFC) 2 / 20

An "easy" question

Letπ :P →M be a fiber bundle with smooth fiber and compact baseM.

Question: When does

vol(s(M))≥vol(M) hold for every sections:M→P?

Possible answer: WhenP→Bis a Riemannian submersion.

vols(M) = Z

M

√

detds^∗ds dvol_M ≥ Z

M

dvol_M ds=id⊕d˜s⇒ds^∗ds=id+d˜s^∗ds˜≥id.

(UFC) 2 / 20

An elementary problem

How about ifP andMare Finsler manifolds?

What is the volume form on a Finsler space?

Definition

Ann-volume density on ann-dimensional vector spaceV is a continuous functionF : ΛⁿV→R≥0such that:

F(λξ) =|λ|F(ξ), F(ξ) =0⇔ξ=0.

A definition of volume is a functorial assignment(V,k · k)→(ΛⁿV,F).

We require that

kTk ≤1⇒ kΛⁿTk ≤1 Remark

If V is oriented, it is enough to take F positively homogenous.

(UFC) 3 / 20

An elementary problem

How about ifP andMare Finsler manifolds?

What is the volume form on a Finsler space?

Definition

Ann-volume density on ann-dimensional vector spaceV is a continuous functionF : ΛⁿV→R≥0such that:

F(λξ) =|λ|F(ξ), F(ξ) =0⇔ξ=0.

A definition of volume is a functorial assignment(V,k · k)→(ΛⁿV,F).

We require that

kTk ≤1⇒ kΛⁿTk ≤1 Remark

If V is oriented, it is enough to take F positively homogenous.

(UFC) 3 / 20

An elementary problem

How about ifP andMare Finsler manifolds?

What is the volume form on a Finsler space?

Definition

Ann-volume density on ann-dimensional vector spaceV is a continuous functionF : ΛⁿV→R≥0such that:

F(λξ) =|λ|F(ξ), F(ξ) =0⇔ξ=0.

A definition of volume is a functorial assignment(V,k · k)→(ΛⁿV,F).

We require that

kTk ≤1⇒ kΛⁿTk ≤1 Remark

If V is oriented, it is enough to take F positively homogenous.

(UFC) 3 / 20

An elementary problem

How about ifP andMare Finsler manifolds?

What is the volume form on a Finsler space?

Definition

Ann-volume density on ann-dimensional vector spaceV is a continuous functionF : ΛⁿV→R≥0such that:

F(λξ) =|λ|F(ξ), F(ξ) =0⇔ξ=0.

A definition of volume is a functorial assignment(V,k · k)→(ΛⁿV,F).

We require that

kTk ≤1⇒ kΛⁿTk ≤1 Remark

If V is oriented, it is enough to take F positively homogenous.

(UFC) 3 / 20

Volume densities/forms

Example

The Busemann-Hausdorff density: (V,k · k)normed space µ_BH(v₁∧. . .∧vn) = n

R

B1(k·k) dv₁∧. . .∧dv_n Example

The Holmes-Thompson density:

µ_HT(v₁∧. . .∧v_n) =⁻¹_n Z

B₁^∗

dv₁^∗∧. . .∧dv_n^∗

M manifold with a volume density.

volM= Z

M

dµ.

(UFC) 4 / 20

Volume densities/forms

Example

The Busemann-Hausdorff density: (V,k · k)normed space µ_BH(v₁∧. . .∧vn) = n

R

B1(k·k) dv₁∧. . .∧dv_n Example

The Holmes-Thompson density:

µ_HT(v₁∧. . .∧v_n) =⁻¹_n Z

B₁^∗

dv₁^∗∧. . .∧dv_n^∗

M manifold with a volume density.

volM= Z

M

dµ.

(UFC) 4 / 20

Jacobians

Definition

Let(V,F),(W,G)be twon-dim vector spaces with volume densities andT :V→W be linear.

J(T) = G(Tv₁∧. . .∧Tv_n)

F(v₁∧. . .∧v_n) =kΛⁿTk_F,G (V,k · k₁)and(V,k · k₂):

J_HT(T) =|detT|vol(B(k · k^∗₂))

vol(B(k · k^∗₁)) J_BH(T) = _n

Hⁿ({x | kTxk ≤1}) Proposition (Change of variables)

Let(M,F),(P,G)be two manifolds with volume densities with degF =degG=dimM and let T :M ,→P be an embedding. Then

Z

T(M)

dG= Z

M

J(dT)dF

(UFC) 5 / 20

Jacobians

Definition

Let(V,F),(W,G)be twon-dim vector spaces with volume densities andT :V→W be linear.

J(T) = G(Tv₁∧. . .∧Tv_n)

F(v₁∧. . .∧v_n) =kΛⁿTk_F,G (V,k · k₁)and(V,k · k₂):

J_HT(T) =|detT|vol(B(k · k^∗₂))

vol(B(k · k^∗₁)) J_BH(T) = _n

Hⁿ({x | kTxk ≤1}) Proposition (Change of variables)

Let(M,F),(P,G)be two manifolds with volume densities with degF =degG=dimM and let T :M ,→P be an embedding. Then

Z

T(M)

dG= Z

M

J(dT)dF

(UFC) 5 / 20

Jacobians

Definition

Let(V,F),(W,G)be twon-dim vector spaces with volume densities andT :V→W be linear.

J(T) = G(Tv₁∧. . .∧Tv_n)

F(v₁∧. . .∧v_n) =kΛⁿTk_F,G (V,k · k₁)and(V,k · k₂):

J_HT(T) =|detT|vol(B(k · k^∗₂))

vol(B(k · k^∗₁)) J_BH(T) = _n

Hⁿ({x | kTxk ≤1}) Proposition (Change of variables)

Let(M,F),(P,G)be two manifolds with volume densities with degF =degG=dimM and let T :M ,→P be an embedding. Then

Z

T(M)

dG= Z

M

J(dT)dF

(UFC) 5 / 20

Jacobians

Proposition

Letπ:P→M be a fiber bundle of compact Finsler manifolds and suppose a definition of volume has been fixed. Ifkdπpk ≤1,∀p ∈P then

vols(M)≥volM, whenever s:M→P is a section.

Remark

Ifπ:P→M is a Riemannian submersion then dπis an orthogonal projection.

(UFC) 6 / 20

Jacobians

Proposition

Letπ:P→M be a fiber bundle of compact Finsler manifolds and suppose a definition of volume has been fixed. Ifkdπpk ≤1,∀p ∈P then

vols(M)≥volM, whenever s:M→P is a section.

Remark

Ifπ:P→M is a Riemannian submersion then dπis an orthogonal projection.

(UFC) 6 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

dimV =n+m,

F : ΛⁿV→R, µ: Λ^n+mV→R⇒ codensity:F/µ: Λ^mV^∗→R,

F/µ(v₁^∗∧. . .∧v_m^∗) = F(u₁∧. . .∧u_n) µ(v₁. . .∧v_m∧u₁∧. . .∧u_n) T :V^n+m→W^m linear,

(V,F, µ)and(W,G), G: Λ^mW→R, G^∗(w₁^∗∧. . .∧w_m^∗) = _G(w ¹

1∧...∧wm),

C(T)_F,µ,G :=J(T^∗)_G^∗_,F/µ= F/µ(T^∗v₁^∗∧. . .∧T^∗v_m^∗) G^∗(v₁^∗∧. . .∧v_m^∗) .

(UFC) 7 / 20

Cojacobians

Example

f :Pⁿ→R,µa volume form onP,

F¯ :TP→R, fiberwise convex, homogeneous F¯^∗ :T^∗P→Rthe dual "norm",F : Λⁿ⁻¹TP→R:

Λⁿ⁻¹T_pP →^∗µ T_p^∗P ^F→^¯∗ R

C(df)_F,µ,dt = ¯F^∗(df)= ¯^! F(∇f),

∇f is the Finslerian gradient:T^∗P\ {0} →TP\ {0}the Legendre transform:

(UFC) 8 / 20

Cojacobians

Example

f :Pⁿ→R,µa volume form onP,

F¯ :TP→R, fiberwise convex, homogeneous F¯^∗ :T^∗P→Rthe dual "norm",F : Λⁿ⁻¹TP→R:

Λⁿ⁻¹T_pP →^∗µ T_p^∗P ^F→^¯∗ R

C(df)_F,µ,dt = ¯F^∗(df)= ¯^! F(∇f),

∇f is the Finslerian gradient:T^∗P\ {0} →TP\ {0}the Legendre transform:

(UFC) 8 / 20

Cojacobians

Example

f :Pⁿ→R,µa volume form onP,

F¯ :TP→R, fiberwise convex, homogeneous F¯^∗ :T^∗P→Rthe dual "norm",F : Λⁿ⁻¹TP→R:

Λⁿ⁻¹T_pP →^∗µ T_p^∗P ^F→^¯∗ R

C(df)_F,µ,dt = ¯F^∗(df)= ¯^! F(∇f),

∇f is the Finslerian gradient:T^∗P\ {0} →TP\ {0}the Legendre transform:

(UFC) 8 / 20

Cojacobians

Example

f :Pⁿ→R,µa volume form onP,

F¯ :TP→R, fiberwise convex, homogeneous F¯^∗ :T^∗P→Rthe dual "norm",F : Λⁿ⁻¹TP→R:

Λⁿ⁻¹T_pP →^∗µ T_p^∗P ^F→^¯∗ R

C(df)_F,µ,dt = ¯F^∗(df)= ¯^! F(∇f),

∇f is the Finslerian gradient:T^∗P\ {0} →TP\ {0}the Legendre transform:

(UFC) 8 / 20

Cojacobians

Example

f :Pⁿ→R,µa volume form onP,

F¯ :TP→R, fiberwise convex, homogeneous F¯^∗ :T^∗P→Rthe dual "norm",F : Λⁿ⁻¹TP→R:

Λⁿ⁻¹T_pP →^∗µ T_p^∗P ^F→^¯∗ R

C(df)_F,µ,dt = ¯F^∗(df)= ¯^! F(∇f),

∇f is the Finslerian gradient:T^∗P\ {0} →TP\ {0}the Legendre transform:

(UFC) 8 / 20

Coarea formula

Theorem

Let f :P^n+m→Mⁿbe a C¹map between smooth manifolds. Let F and µbe volume densities on P andλan n-density on M. Let A⊂P be a measurable set. Then:

Z

A

C(df)F,µ,λ dµ= Z

M

vol_F(A∩π⁻¹(y))dλ(y).

(UFC) 9 / 20

Proof(sketch)

First reduce to the case off :R^n+m→Rⁿ. Lemma

Let V,W,Z three vector spaces, A:V→W , T :W→Z . (a) IfdimV =n,dimW =n,µ∈Dn(V),λ∈Dn(W):

J(A;µ, λ) =C(A;µ^∗, λ^∗).

(b) IfdimV =n,µ∈Dn(V), G∈Dn(W), H ∈Dn(Z):

J(T ◦A;µ,H) =J(A;µ,G)·J(T

ImA;G,H).

(c) IfdimZ =n,λ∈Dn(Z), F/µ∈Dn(V^∗), G/ν ∈Dn(W^∗):

C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A^∗

_ImT∗;G/ν,F/µ).

(d) The same conditions as in(c)with A isomorphism and T surjective:

J(A

KerT◦A;F,G)·C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A;µ, ν).

(UFC) 10 / 20

Proof(sketch)

First reduce to the case off :R^n+m→Rⁿ. Lemma

Let V,W,Z three vector spaces, A:V→W , T :W→Z . (a) IfdimV =n,dimW =n,µ∈Dn(V),λ∈Dn(W):

J(A;µ, λ) =C(A;µ^∗, λ^∗).

(b) IfdimV =n,µ∈Dn(V), G∈Dn(W), H ∈Dn(Z):

J(T ◦A;µ,H) =J(A;µ,G)·J(T

ImA;G,H).

(c) IfdimZ =n,λ∈Dn(Z), F/µ∈Dn(V^∗), G/ν ∈Dn(W^∗):

C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A^∗

_ImT∗;G/ν,F/µ).

(d) The same conditions as in(c)with A isomorphism and T surjective:

J(A

KerT◦A;F,G)·C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A;µ, ν).

(UFC) 10 / 20

Proof(sketch)

First reduce to the case off :R^n+m→Rⁿ. Lemma

Let V,W,Z three vector spaces, A:V→W , T :W→Z . (a) IfdimV =n,dimW =n,µ∈Dn(V),λ∈Dn(W):

J(A;µ, λ) =C(A;µ^∗, λ^∗).

(b) IfdimV =n,µ∈Dn(V), G∈Dn(W), H ∈Dn(Z):

J(T ◦A;µ,H) =J(A;µ,G)·J(T

ImA;G,H).

(c) IfdimZ =n,λ∈Dn(Z), F/µ∈Dn(V^∗), G/ν ∈Dn(W^∗):

C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A^∗

_ImT∗;G/ν,F/µ).

(d) The same conditions as in(c)with A isomorphism and T surjective:

J(A

KerT◦A;F,G)·C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A;µ, ν).

(UFC) 10 / 20

Proof(sketch)

First reduce to the case off :R^n+m→Rⁿ. Lemma

Let V,W,Z three vector spaces, A:V→W , T :W→Z . (a) IfdimV =n,dimW =n,µ∈Dn(V),λ∈Dn(W):

J(A;µ, λ) =C(A;µ^∗, λ^∗).

(b) IfdimV =n,µ∈Dn(V), G∈Dn(W), H ∈Dn(Z):

J(T ◦A;µ,H) =J(A;µ,G)·J(T

ImA;G,H).

(c) IfdimZ =n,λ∈Dn(Z), F/µ∈Dn(V^∗), G/ν ∈Dn(W^∗):

C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A^∗

_ImT∗;G/ν,F/µ).

(d) The same conditions as in(c)with A isomorphism and T surjective:

J(A

KerT◦A;F,G)·C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A;µ, ν).

(UFC) 10 / 20

Proof(sketch)

First reduce to the case off :R^n+m→Rⁿ. Lemma

Let V,W,Z three vector spaces, A:V→W , T :W→Z . (a) IfdimV =n,dimW =n,µ∈Dn(V),λ∈Dn(W):

J(A;µ, λ) =C(A;µ^∗, λ^∗).

(b) IfdimV =n,µ∈Dn(V), G∈Dn(W), H ∈Dn(Z):

J(T ◦A;µ,H) =J(A;µ,G)·J(T

ImA;G,H).

(c) IfdimZ =n,λ∈Dn(Z), F/µ∈Dn(V^∗), G/ν ∈Dn(W^∗):

C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A^∗

_ImT∗;G/ν,F/µ).

(d) The same conditions as in(c)with A isomorphism and T surjective:

J(A

KerT◦A;F,G)·C(T ◦A;λ^∗,F/µ) =C(T;λ^∗,G/ν)·J(A;µ, ν).

(UFC) 10 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

The isoperimetric problem on normed vector spaces

V a normed vector space andµvolume form, translation invariant;

F : Λⁿ⁻¹V→Ra norm with unit ballB

the isoperimetrix: I:=ι_µ(B)^∗ ⊂V (ι_µ: Λⁿ⁻¹V→V^∗)

Alvarez-Paiva and Thompson:Isolves the isoperimetric problem for convex bodies.

WhenI=B₁(k · k)?

Proposition

Precisely when F/µ=k · k^∗ ⇔C(f;F/µ,dt) =kfk^∗,∀f ∈V^∗

⇔µ(w∧v₁∧. . .∧vn−1) =k[w]k ·F(v₁∧. . .∧vn−1).

(UFC) 11 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

Anisotropy

Vⁿvector space withµvolume form, translation invariant.

F¯ :V→Ra Minkowski norm with unit ballW_F (Wulff body).

the area density: F : Λⁿ⁻¹V→R.

the anisotropic area functional: Σ⊂V oriented hypersurface F(Σ) =R

Σ dF.

ifh·,·ionV, leth:V→Rbe the support function ofW_F. h'F¯^∗ underV 'V^∗

R

Σ dF =R

Σh(ν)dHⁿ⁻¹

J(id;F;Hⁿ⁻¹) = F(v₁∧. . .∧vn−1)

Hⁿ⁻¹(v₁∧. . .v_n−1) = F(v₁∧. . .∧vn−1)

Ω(ν∧v₁∧. . .v_n−1) = ¯F^∗(ν^∗)

(UFC) 12 / 20

The anisotropic Sobolev inequality

Theorem (Gromov) Let f ∈C_c^∞(V;R).

Z

V

h(−gradf_x)dµ≥nvol(W_F)¹ⁿ Z

V

|f|^{n−1n n−1}_n

Gromov proved this first using ideas of optimal transport.

We use the classical idea of Federer-Fleming: reduce it to an anisotropic isoperimetric inequality:

Dsmooth domain Area_F(∂D) :=

Z

∂D

dF ≥nvol(W_F)¹ⁿvol(D)ⁿ⁻¹ⁿ . The Brunn-Minkowski inequalityD,W ⊂V:

vol¹ⁿ(D+tW)≥vol¹ⁿ(D) +vol¹ⁿ(tW)

(UFC) 13 / 20

The anisotropic Sobolev inequality

Theorem (Gromov) Let f ∈C_c^∞(V;R).

Z

V

h(−gradf_x)dµ≥nvol(W_F)¹ⁿ Z

V

|f|^{n−1n n−1}_n

Gromov proved this first using ideas of optimal transport.

We use the classical idea of Federer-Fleming: reduce it to an anisotropic isoperimetric inequality:

Dsmooth domain Area_F(∂D) :=

Z

∂D

dF ≥nvol(W_F)¹ⁿvol(D)ⁿ⁻¹ⁿ . The Brunn-Minkowski inequalityD,W ⊂V:

vol¹ⁿ(D+tW)≥vol¹ⁿ(D) +vol¹ⁿ(tW)

(UFC) 13 / 20

The anisotropic Sobolev inequality

Theorem (Gromov) Let f ∈C_c^∞(V;R).

Z

V

h(−gradf_x)dµ≥nvol(W_F)¹ⁿ Z

V

|f|^{n−1n n−1}_n

Gromov proved this first using ideas of optimal transport.

We use the classical idea of Federer-Fleming: reduce it to an anisotropic isoperimetric inequality:

Dsmooth domain Area_F(∂D) :=

Z

∂D

dF ≥nvol(W_F)¹ⁿvol(D)ⁿ⁻¹ⁿ . The Brunn-Minkowski inequalityD,W ⊂V:

vol¹ⁿ(D+tW)≥vol¹ⁿ(D) +vol¹ⁿ(tW)

(UFC) 13 / 20

The anisotropic Sobolev inequality

Theorem (Gromov) Let f ∈C_c^∞(V;R).

Z

V

h(−gradf_x)dµ≥nvol(W_F)¹ⁿ Z

V

|f|^{n−1n n−1}_n

Gromov proved this first using ideas of optimal transport.

We use the classical idea of Federer-Fleming: reduce it to an anisotropic isoperimetric inequality:

Dsmooth domain Area_F(∂D) :=

Z

∂D

dF ≥nvol(W_F)¹ⁿvol(D)ⁿ⁻¹ⁿ . The Brunn-Minkowski inequalityD,W ⊂V:

vol¹ⁿ(D+tW)≥vol¹ⁿ(D) +vol¹ⁿ(tW)

(UFC) 13 / 20

The anisotropic Sobolev inequality

Theorem (Gromov) Let f ∈C_c^∞(V;R).

Z

V

h(−gradf_x)dµ≥nvol(W_F)¹ⁿ Z

V

|f|^{n−1n n−1}_n

Gromov proved this first using ideas of optimal transport.

We use the classical idea of Federer-Fleming: reduce it to an anisotropic isoperimetric inequality:

Dsmooth domain Area_F(∂D) :=

Z

∂D

dF ≥nvol(W_F)¹ⁿvol(D)ⁿ⁻¹ⁿ . The Brunn-Minkowski inequalityD,W ⊂V:

vol¹ⁿ(D+tW)≥vol¹ⁿ(D) +vol¹ⁿ(tW)

(UFC) 13 / 20

The outer anisotropic Minkowski content

Theorem

D bounded, smooth domain (only need C²for the proof)

tlim→0

vol(D+tW_F)−vol(D)

t =Area_F(∂D)

Da set of finite perimeter−−>Chambolle, Lisini and Lussardi.

(UFC) 14 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

First assumeh_WF is smooth (in particularW_F is strictly convex).

Letn:∂D→∂W_F the oriented anisotropic Gauss map.

∂D→^ν S^n−1grad−→^hWF ∂W_F hencenisC¹at least.

φ: [0,t]×∂D→V, (s,b)→b+sn(b).

fort small,φis an oriented diffeo ontoD˜t := ¯D+tW_F \D(use that theF¯ geodesics are lines).

volD˜_t =R

[0,t]×∂DJ(dφ;ds×F, µ)ds⊗dF.

|J_φ(s,b)(dφ;ds×dF, µ)−1| ≤Ct fors ≤tandt small.

J_φ(s,b)= ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1)

(UFC) 15 / 20

The outer anisotropic Minkowski content

J_φ(0,b)=1 due to the multiplication property:

µ(n∧v₁∧. . .∧v_n−1) = ¯F(n)F(v₁∧. . .∧v_n−1).

φ(s,b) =b+sn(b)⇒ ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1) polynomial ins.

1 t

R

[0,t]×∂D1−J(dφ;ds×F, µ)ds⊗dF ≤ R

[0,t]×∂DM ds⊗dF→0.

forh_WF not smooth use an approximation argument due to Schneider.

to prove the anisotropic Sobolev use the regular level sets of

|f|⁻¹(t)and the coarea formula with

C(d_x|f|;dt,F/µ) =h_WF(−grad_xf)

(UFC) 16 / 20

The outer anisotropic Minkowski content

J_φ(0,b)=1 due to the multiplication property:

µ(n∧v₁∧. . .∧v_n−1) = ¯F(n)F(v₁∧. . .∧v_n−1).

φ(s,b) =b+sn(b)⇒ ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1) polynomial ins.

1 t

R

[0,t]×∂D1−J(dφ;ds×F, µ)ds⊗dF ≤ R

[0,t]×∂DM ds⊗dF→0.

forh_WF not smooth use an approximation argument due to Schneider.

to prove the anisotropic Sobolev use the regular level sets of

|f|⁻¹(t)and the coarea formula with

C(d_x|f|;dt,F/µ) =h_WF(−grad_xf)

(UFC) 16 / 20

The outer anisotropic Minkowski content

J_φ(0,b)=1 due to the multiplication property:

µ(n∧v₁∧. . .∧v_n−1) = ¯F(n)F(v₁∧. . .∧v_n−1).

φ(s,b) =b+sn(b)⇒ ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1) polynomial ins.

1 t

R

[0,t]×∂D1−J(dφ;ds×F, µ)ds⊗dF ≤ R

[0,t]×∂DM ds⊗dF→0.

forh_WF not smooth use an approximation argument due to Schneider.

to prove the anisotropic Sobolev use the regular level sets of

|f|⁻¹(t)and the coarea formula with

C(d_x|f|;dt,F/µ) =h_WF(−grad_xf)

(UFC) 16 / 20

The outer anisotropic Minkowski content

J_φ(0,b)=1 due to the multiplication property:

µ(n∧v₁∧. . .∧v_n−1) = ¯F(n)F(v₁∧. . .∧v_n−1).

φ(s,b) =b+sn(b)⇒ ^µ(nb∧db_F^φs_(v^{(v1)∧...∧dbφs(vn−1))}

1∧...∧v_n−1) polynomial ins.

1 t

R

[0,t]×∂D1−J(dφ;ds×F, µ)ds⊗dF ≤ R

[0,t]×∂DM ds⊗dF→0.

forh_WF not smooth use an approximation argument due to Schneider.

to prove the anisotropic Sobolev use the regular level sets of

|f|⁻¹(t)and the coarea formula with

C(d_x|f|;dt,F/µ) =h_WF(−grad_xf)

(UFC) 16 / 20