Presentation Jun O'Hara pisa

. . . . . .

.

. . . .

.

.

Renormalization of r

-potentials and its

applications

Jun O’Hara (Tokyo Metropolitan University)

06/2011

Centro di Ricerca Matematica, Ennio De Giorgi, Pisa

. . . . . .

.. Motivations

Problem(PISA) : Let Ω be a triangular park. Where should we put a street light?

PISA’answer : the circumcenter (the point that makes the darkest place brightest if Ω is not obtuse).

Shibata(Fukuoka)’s answer : a point that maximizes the

“total brightness” .Definition

. .

... .

.

.

A point is an illuminating center of Ω if it maximizes Ω∋ x 7→◦

∫

Ω

dy

|x − y|²

integrated after the divergence of the integrand is cut-off

. . . . . .

.. Motivations

Problem(PISA) : Let Ω be a triangular park. Where should we put a street light?

PISA’answer : the circumcenter (the point that makes the darkest place brightest if Ω is not obtuse).

Shibata(Fukuoka)’s answer : a point that maximizes the

“total brightness” .Definition

. .

... .

.

.

A point is an illuminating center of Ω if it maximizes Ω∋ x 7→◦

∫

Ω

dy

|x − y|²

integrated after the divergence of the integrand is cut-off

. . . . . .

.. Motivations

Problem(PISA) : Let Ω be a triangular park. Where should we put a street light?

PISA’answer : the circumcenter (the point that makes the darkest place brightest if Ω is not obtuse).

Shibata(Fukuoka)’s answer : a point that maximizes the

“total brightness” .Definition

. .

... .

.

.

A point is an illuminating center of Ω if it maximizes Ω∋ x 7→◦

∫

Ω

dy

|x − y|²

integrated after the divergence of the integrand is cut-off

. . . . . .

.. Motivations

Problem(PISA) : Let Ω be a triangular park. Where should we put a street light?

PISA’answer : the circumcenter (the point that makes the darkest place brightest if Ω is not obtuse).

Shibata(Fukuoka)’s answer : a point that maximizes the

“total brightness” .Definition

. .

... .

.

.

A point is an illuminating center of Ω if it maximizes Ω∋ x 7→◦

∫

Ω

dy

|x − y|²

integrated after the divergence of the integrand is cut-off

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Outline of the first half

The renormalization used to make a family of energy functionals of knots (O ’92)

⇝ 1-parameter family of renormalized potentials

⇝ 1-parameter family of points that maximizes (or minimizes) the potentials (“generalized centers”)

One limit→ a min-max point (e.g. circumcenter of a non-obtuse triangle)

This family∋the center of mass This family∋the illuminating center

The other limit → a max-min point (e.g. incenter of a triangle) Generalization of “dual volume” in convex geometry

Uniqueness of the “centers”

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. Definitions

R^m ⊃ Ω (m ≥ 2) : compact, ∂Ω : piecewise C¹ .Definition

. .

... .

.

.

If α > 0 or w ̸∈ Ω

V_Ω^(α)(w) :=

∫

Ω

|w − z|^α−mdz It is called the Riesz potential when 0 < α < m If α ≤ 0 and w ∈ Ω

V_Ω^(α)(w) := lim

ε_→0

(∫

Ω\Bε(w)

|w − z|^α−mdz+^A(S

m₋₁

)

α ^ε

α

) , where A(S^m−1) is the volume of the unit sphere S^m−1 Put _α¹r^α= log r (r > 0) if α = 0 in the formulae (although log r = lima→0¹_a(r^a− 1) in fact) When m = 2 and α = −2, Auckly-Sadun.

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0

Proof:

∫

R^m_\Bε(w)

|w − z|^α−mdz= −^A(S

m−1₎

α ^{· ε}

α

V_Ω^(α)(w)

= lim

ε→0

(∫

Ω\Bε(w)

|w − z|^α−mdz−

∫

R^m_\Bε(w)

|w − z|^α−mdz )

= −

∫

Ω^c

|w − z|^α−mdz .

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0

Proof:

∫

R^m_\Bε(w)

|w − z|^α−mdz= −^A(S

m−1₎

α ^{· ε}

α

V_Ω^(α)(w)

= lim

ε→0

(∫

Ω\Bε(w)

|w − z|^α−mdz−

∫

R^m_\Bε(w)

|w − z|^α−mdz )

= −

∫

Ω^c

|w − z|^α−mdz .

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0

Proof:

∫

R^m_\Bε(w)

|w − z|^α−mdz= −^A(S

m−1₎

α ^{· ε}

α

V_Ω^(α)(w)

= lim

ε→0

(∫

Ω\Bε(w)

|w − z|^α−mdz−

∫

R^m_\Bε(w)

|w − z|^α−mdz )

= −

∫

Ω^c

|w − z|^α−mdz .

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

Ω: conex, w∈Ω ⇒ V^◦ _Ω^{(α)(w) = 1} α

∫

S^m−1

(ρ(v))^α dσ(v), where ρ : S^m−1 ∋ v 7→ sup{c > 0 | w + cv ∈ Ω} ∈ R+^.

ρ is called the radial function.

∫

S^m−1(ρ(v))

α dσ(v) is called the dual volume of order α (Lutwak 75,88).

A similar formula holds for general regions (after we divide Ω into convex parts)

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

Ω: conex, w∈Ω ⇒ V^◦ _Ω^{(α)(w) = 1} α

∫

S^m−1

(ρ(v))^α dσ(v), where ρ : S^m−1 ∋ v 7→ sup{c > 0 | w + cv ∈ Ω} ∈ R+^.

ρ is called the radial function.

∫

S^m−1(ρ(v))

α dσ(v) is called the dual volume of order α (Lutwak 75,88).

A similar formula holds for general regions (after we divide Ω into convex parts)

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

Ω: conex, w∈Ω ⇒ V^◦ _Ω^{(α)(w) = 1} α

∫

S^m−1

(ρ(v))^α dσ(v), where ρ : S^m−1 ∋ v 7→ sup{c > 0 | w + cv ∈ Ω} ∈ R+^.

ρ is called the radial function.

∫

S^m−1(ρ(v))

α dσ(v) is called the dual volume of order α (Lutwak 75,88).

A similar formula holds for general regions (after we divide Ω into convex parts)

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈_{Ω ⇒ V}^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

Ω: conex, w∈Ω ⇒ V^◦ _Ω^{(α)(w) = 1} α

∫

S^m−1

(ρ(v))^α dσ(v), where ρ : S^m−1 ∋ v 7→ sup{c > 0 | w + cv ∈ Ω} ∈ R+^.

ρ is called the radial function.

∫

S^m−1(ρ(v))

α dσ(v) is called the dual volume of order α (Lutwak 75,88).

A similar formula holds for general regions (after we divide Ω into convex parts)

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈Ω ⇒ V^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

α≤ 0 ⇒

{ _◦

Ω∋ w → ∂Ω ⇒ V_Ω^{(α)(w) → −∞} Ω^c ∋ w → ∂Ω ⇒ V_Ω^(α)(w) → +∞

. . . . . .

.. ^Properties

.Proposition .

.

... .

.

.

α <0, w ∈Ω ⇒ V^◦ _Ω^(α)(w) = −

∫

Ω^c

|w − z|^α−mdz <0 .Proposition

. .

... .

.

.

α≤ 0 ⇒

{ _◦

Ω∋ w → ∂Ω ⇒ V_Ω^{(α)(w) → −∞} Ω^c ∋ w → ∂Ω ⇒ V_Ω^(α)(w) → +∞

. . . . . .

.. ^r

^-centers

.Definition .

.

... .

.

.

An r^α−m-centerof Ω is a point that attains the extreme value of V_Ω^(α), to be precise, a point that

minimizes V_Ω^(α) if α > m, maximizes V_Ω^(α) if 0 < α < m, maximizes

∫

Ω

− log |w − z|dz if α = m, maximizes V_Ω^(α)◦

Ω ^{if α ≤ 0.}

Remark that V_Ω^(m)(w) ≡ Vol(Ω) when α = m

. . . . . .

.. ^r

^-centers

.Definition .

.

... .

.

.

An r^α−m-centerof Ω is a point that attains the extreme value of V_Ω^(α), to be precise, a point that

minimizes V_Ω^(α) if α > m, maximizes V_Ω^(α) if 0 < α < m, maximizes

∫

Ω

− log |w − z|dz if α = m, maximizes V_Ω^(α)◦

Ω ^{if α ≤ 0.}

Remark that V_Ω^(m)(w) ≡ Vol(Ω) when α = m

. . . . . .

.. ^r

^-centers

.Definition .

.

... .

.

.

An r^α−m-centerof Ω is a point that attains the extreme value of V_Ω^(α), to be precise, a point that

minimizes V_Ω^(α) if α > m, maximizes V_Ω^(α) if 0 < α < m, maximizes

∫

Ω

− log |w − z|dz if α = m, maximizes V_Ω^(α)◦

Ω ^{if α ≤ 0.}

Remark that V_Ω^(m)(w) ≡ Vol(Ω) when α = m

. . . . . .

.. ^r

^-centers

.Definition .

.

... .

.

.

An r^α−m-centerof Ω is a point that attains the extreme value of V_Ω^(α), to be precise, a point that

minimizes V_Ω^(α) if α > m, maximizes V_Ω^(α) if 0 < α < m, maximizes

∫

Ω

− log |w − z|dz if α = m, maximizes V_Ω^(α)◦

Ω ^{if α ≤ 0.}

Remark that V_Ω^(m)(w) ≡ Vol(Ω) when α = m

. . . . . .

.. ^r

^-centers

.Definition .

.

... .

.

.

An r^α−m-centerof Ω is a point that attains the extreme value of V_Ω^(α), to be precise, a point that

minimizes V_Ω^(α) if α > m, maximizes V_Ω^(α) if 0 < α < m, maximizes

∫

Ω

− log |w − z|dz if α = m, maximizes V_Ω^(α)◦

Ω ^{if α ≤ 0.}

Remark that V_Ω^(m)(w) ≡ Vol(Ω) when α = m

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. ^Examples

The center of mass wG of Ω is given by wG =

∫

Ω^zdz

∫

Ω^1dz

.

⇔

∫

Ω

(wG− z) dz = 0.

⇔ wG is a unique critical point of

∫

Ω

|w − z|²dz . It is a unique r²-center (α = m + 2)

The illuminating center of a planar domain (Shibata) is an r⁻²-center (m = 2, α = 0).

When m = 3, an r⁻¹-center (α = 2) is

a “gravitational center” or a “point with highest voltage” (Herbur-Moszy´nska-Peradz´ynski)

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Existence and uniqueness of r

-centers

.Theorem .

.

... .

. .Any compact region has an r^α−m-center for any α.

r^a-center is not necessarily unique. .Example

. .

... .

.

.

Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R).

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. Uniqueness of r

-centers

.Theorem .

.

... .

.

.

A compact region Ω has a unique r^α−m-center if either α≥ m + 1,

α≤ 1 and Ω is convex. .Proof.

. .

... .

.

.

∂²V_Ω^(α)

∂x_j^{2 >0 on R}

m if α ≥ m + 1

∂²V_Ω^(α)

∂x_j^{2 <0 on}

Ω if α ≤ 1 and Ω is convex◦

.Conjecture .

.

... .

. .A convex region has a uniqe r^α−m-center for any α.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R²y^min∈Ω^{c|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R²y^min∈Ω^{c|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R²y^min∈Ω^{c|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R²y^min∈Ω^{c|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R²y^min∈Ω^{c|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. ^{An r}

-center of a triangle

An r^α−2-center of a triangle ∆ is a point x that gives min

x∈R^2maxy∈Ω|x − y| (α → +∞). It is the circumcenter if ∆ is not obtuse

the barycenter = r²-center (α = 4)

the illuminating center = r^α−2-center (α = 0) the incenter = a point x that gives max

x∈R^{2ymin∈Ωc|x − y|}

(α → −∞)

In general, a min-max point is uniquely determined, but a max-min point is not. The latter is contained in the so-called medial axis.

. . . . . .

.. Boundary integral expression and Laplacians

.Theorem .

.

... .

.

.

If w ̸∈ ∂Ω then for any α V_Ω^(α)(w) = ¹

α

∫

∂Ω

|w − z|^α−m(z − w) · n dσ(z)

We can compute the derivatives of V_Ω^(α) .Proposition

. .

... .

.

.

α >2 or (α ̸= 2 and w ̸∈ ∂Ω) ⇒

∆V_Ω^(α)(w) = (α − 2)(α − m)V_Ω^(α−2)(w)

. . . . . .

.. Boundary integral expression and Laplacians

.Theorem .

.

... .

.

.

If w ̸∈ ∂Ω then for any α V_Ω^(α)(w) = ¹

α

∫

∂Ω

|w − z|^α−m(z − w) · n dσ(z)

We can compute the derivatives of V_Ω^(α) .Proposition

. .

... .

.

.

α >2 or (α ̸= 2 and w ̸∈ ∂Ω) ⇒

∆V_Ω^(α)(w) = (α − 2)(α − m)V_Ω^(α−2)(w)

. . . . . .

.. Boundary integral expression and Laplacians

.Theorem .

.

... .

.

.

If w ̸∈ ∂Ω then for any α V_Ω^(α)(w) = ¹

α

∫

∂Ω

|w − z|^α−m(z − w) · n dσ(z)

We can compute the derivatives of V_Ω^(α) .Proposition

. .

... .

.

.

α >2 or (α ̸= 2 and w ̸∈ ∂Ω) ⇒

∆V_Ω^(α)(w) = (α − 2)(α − m)V_Ω^(α−2)(w)

. . . . . .

.. Boundary integral expression and Laplacians

.Theorem .

.

... .

.

.

If w ̸∈ ∂Ω then for any α V_Ω^(α)(w) = ¹

α

∫

∂Ω

|w − z|^α−m(z − w) · n dσ(z)

We can compute the derivatives of V_Ω^(α) .Proposition

. .

... .

.

.

α >2 or (α ̸= 2 and w ̸∈ ∂Ω) ⇒

∆V_Ω^(α)(w) = (α − 2)(α − m)V_Ω^(α−2)(w)

. . . . . .

.. Minimal unfolded region

.Definition .

.

... .

.

.

M_v := sup

x∈Ω

x· v (v ∈ S^m−1),

Ω⁺_v,b:= Ω ∩ {x ∈ R^m| x · v > b} (b ∈ R), Reflv,b:= a reflection in {x ∈ R^m| x · v = b}.

l_v := inf{a | a ≤ Mv^,Reflv,b^(Ω+_v,b) ⊂ Ω (a ≤ ∀b ≤ M_v)}. Define the minimal unfolded region of Ω by

U f(Ω) = ^∩

v∈S^m−1

{x ∈ R^m| x · v ≤ lv} .

. . . . . .

.. Examples of minimal unfolded regions

U f(Ω) ̸⊂ Ω

. . . . . .

.. Examples of minimal unfolded regions

U f(Ω) ̸⊂ Ω

. . . . . .

.. Examples of minimal unfolded regions

U f(Ω) ̸⊂ Ω

. . . . . .

.. Centers in minimal unfolded regions

The integrands of our potentials and their derivatives are quite symmetric.

The moving plane method (Gidas-Ni-Nirenberg) implies .Theorem

. .

... .

. .Any r^α−m-center of Ω is in the minimal unfolded region of Ω.