Whitney Extension Theorems for convex functions ofthe classes C1 and C1,ω.

Daniel Azagra and Carlos Mudarra

The 9th Whitney Problems Workshop

Haifa, May 30, 2015

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 1 / 35

Previous results concerning extensions of smooth convex functions

Previous results concerning extensions of smooth convex functions

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 2 / 35

Previous results concerning extensions of smooth convex functions

M. Ghomi (2002) and M. Yan (2013) have considered the followingquestion: if f : C ⊂ Rn → R is convex and Cm, when can we find a convexfunction F ∈ Cm(Rn) such that F = f on C?

A partial solution to this question, provided by these authors is: always,provided that m ≥ 2, C is a compact convex body and D2f > 0 on ∂C.

Observe that positive definiteness of D2f on ∂C is a very strong condition,far from being necessary.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 3 / 35

Previous results concerning extensions of smooth convex functions

M. Ghomi (2002) and M. Yan (2013) have considered the followingquestion: if f : C ⊂ Rn → R is convex and Cm, when can we find a convexfunction F ∈ Cm(Rn) such that F = f on C?

A partial solution to this question, provided by these authors is: always,provided that m ≥ 2, C is a compact convex body and D2f > 0 on ∂C.

Observe that positive definiteness of D2f on ∂C is a very strong condition,far from being necessary.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 3 / 35

Previous results concerning extensions of smooth convex functions

M. Ghomi (2002) and M. Yan (2013) have considered the followingquestion: if f : C ⊂ Rn → R is convex and Cm, when can we find a convexfunction F ∈ Cm(Rn) such that F = f on C?

A partial solution to this question, provided by these authors is: always,provided that m ≥ 2, C is a compact convex body and D2f > 0 on ∂C.

Observe that positive definiteness of D2f on ∂C is a very strong condition,far from being necessary.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 3 / 35

Previous results concerning extensions of smooth convex functions

A related result of Ghomi’s, concerning interpolation of smoothsubmanifolds in Rn by boundaries of smooth ovaloids (that is stronglyconvex bodies), is as follows: say that a C2 submanifold M ⊂ Rn isstrongly convex if through every point of M there passes a nonsingularsupport hyperplane (i.e. a support hyperplane with contact of order one)with respect to which M lies strictly on one side. Then one has:

Theorem (M. Ghomi, J. Diff. Geom., 2002)Let m ≥ 2 and M ⊂ Rn be a compact embedded submanifold of class Cm,possibly with boundary. Then M is contained in a Cm smooth ovaloid ifand only if M is strongly convex. Moreover, for every smooth field ofnonsingular support hyperplanes along M, there is a smooth ovaloid whichcontains M and is tangent to the given field.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 4 / 35

Previous results concerning extensions of smooth convex functions

A related result of Ghomi’s, concerning interpolation of smoothsubmanifolds in Rn by boundaries of smooth ovaloids (that is stronglyconvex bodies), is as follows: say that a C2 submanifold M ⊂ Rn isstrongly convex if through every point of M there passes a nonsingularsupport hyperplane (i.e. a support hyperplane with contact of order one)with respect to which M lies strictly on one side. Then one has:

Theorem (M. Ghomi, J. Diff. Geom., 2002)Let m ≥ 2 and M ⊂ Rn be a compact embedded submanifold of class Cm,possibly with boundary. Then M is contained in a Cm smooth ovaloid ifand only if M is strongly convex. Moreover, for every smooth field ofnonsingular support hyperplanes along M, there is a smooth ovaloid whichcontains M and is tangent to the given field.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 4 / 35

Previous results concerning extensions of smooth convex functions

It’s tempting to formulate these kind of problems for more general sets (forinstance arbitrary compact sets K instead of manifolds M ⊂ Rn) and moregeneral functions (for instance, f : C→ R with C not necessarily convex).That’s the starting point of our paper.

In particular, we were interested in obtaining versions of the classicalWhitney extension theorem for the class of convex functions. A naturalquestion is:

Given C ⊂ Rn and f : C→ R, G : C→ Rn continuous, what conditionsare necessary and sufficient to ensure the existence of a convex functionF ∈ C1(Rn) (resp. F ∈ C1,ω(Rn)) such that F = f on C and∇F = G onC?

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 5 / 35

Previous results concerning extensions of smooth convex functions

It’s tempting to formulate these kind of problems for more general sets (forinstance arbitrary compact sets K instead of manifolds M ⊂ Rn) and moregeneral functions (for instance, f : C→ R with C not necessarily convex).That’s the starting point of our paper.

In particular, we were interested in obtaining versions of the classicalWhitney extension theorem for the class of convex functions. A naturalquestion is:

Given C ⊂ Rn and f : C→ R, G : C→ Rn continuous, what conditionsare necessary and sufficient to ensure the existence of a convex functionF ∈ C1(Rn) (resp. F ∈ C1,ω(Rn)) such that F = f on C and∇F = G onC?

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 5 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

Classical Whitney Extension Theorems for C1 and C1,ω

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 6 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

Let’s start by restating the classical Whitney extension theorem for C1 inthe form that we will be referring to.

Theorem (Restatement of Whitney’s extension theorem for C1)Let C ⊂ Rn be closed. A necessary and sufficient condition, for a functionf : C→ R, and a continuous mapping G : C→ Rn, to admit a C1

extension F to all of Rn with∇F = G on C, is that

lim|z−y|→0+

f (z)− f (y)− 〈G(y), z− y〉|z− y|

= 0, (W1)

uniformly on compact subsets of C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 7 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

By a modulus of continuity ω we’ll understand a concave, strictlyincreasing function ω : [0,∞)→ [0,∞) such that ω(0+) = 0.

In particular, ω has an inverse ω−1 : [0, β)→ [0,∞) which is convex andstrictly increasing, where β > 0 may be finite or infinite (according towhether ω is bounded or unbounded).

It is well-known that for every uniformly continuous function f : X → Ybetween two metric spaces there exists a modulus of continuity ω such thatdY(f (x), f (z)) ≤ ω (dX(x, z)) for every x, z ∈ X.Abusing terminology, we will say that a mapping G : X → Y has modulusof continuity ω (or that G is ω-continuous) if there exists M ≥ 0 such that

dY (G(x),G(y)) ≤ Mω (dX(x, y))

for all x, y ∈ X.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 8 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

By a modulus of continuity ω we’ll understand a concave, strictlyincreasing function ω : [0,∞)→ [0,∞) such that ω(0+) = 0.

In particular, ω has an inverse ω−1 : [0, β)→ [0,∞) which is convex andstrictly increasing, where β > 0 may be finite or infinite (according towhether ω is bounded or unbounded).

It is well-known that for every uniformly continuous function f : X → Ybetween two metric spaces there exists a modulus of continuity ω such thatdY(f (x), f (z)) ≤ ω (dX(x, z)) for every x, z ∈ X.

Abusing terminology, we will say that a mapping G : X → Y has modulusof continuity ω (or that G is ω-continuous) if there exists M ≥ 0 such that

dY (G(x),G(y)) ≤ Mω (dX(x, y))

for all x, y ∈ X.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 8 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

By a modulus of continuity ω we’ll understand a concave, strictlyincreasing function ω : [0,∞)→ [0,∞) such that ω(0+) = 0.

In particular, ω has an inverse ω−1 : [0, β)→ [0,∞) which is convex andstrictly increasing, where β > 0 may be finite or infinite (according towhether ω is bounded or unbounded).

It is well-known that for every uniformly continuous function f : X → Ybetween two metric spaces there exists a modulus of continuity ω such thatdY(f (x), f (z)) ≤ ω (dX(x, z)) for every x, z ∈ X.Abusing terminology, we will say that a mapping G : X → Y has modulusof continuity ω (or that G is ω-continuous) if there exists M ≥ 0 such that

dY (G(x),G(y)) ≤ Mω (dX(x, y))

for all x, y ∈ X.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 8 / 35

Classical Whitney Extension Theorems for C1 and C1,ω

Theorem (Restatement of Glaeser’s version of the classical Whitneyextension theorem, case C1,ω)Let C be a subset of Rn, f : C→ R a function, and G : C→ Rn auniformly continuous mapping, with modulus of continuity ω. Assume thatthe pair (f ,G) satisfies

|f (x)− f (y)− 〈G(y), x− y〉| ≤ M|x− y|ω (|x− y|) (W1,ω)

for every x, y ∈ C. Then there exists a function F ∈ C1,ω(Rn) such thatF = f and∇F = G on C.

Moreover,

‖F‖C1,ω(Rn) ≤ k(n)

(supy∈C|f (y)|+ sup

x,y∈C,x 6=y

|G(x)− G(y)|ω (|x− y|)

),

where k(n) only depends on the dimension n.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 9 / 35

C1,ω convex extensions of functions

C1,ω convex extensions of functions

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 10 / 35

C1,ω convex extensions of functions

For a mapping G : C ⊂ Rn → Rn we will denote

M(G,C) := supx,y∈C, x 6=y

|G(x)− G(y)|ω(|x− y|)

. (3.1)

We will say that f and G satisfy the property (CW1,ω) on C if either G isconstant, or else 0 < M(G,C) <∞ and

f (x)− f (y)− 〈G(y), x− y〉 ≥ 12|G(x)− G(y)|ω−1

(1

2M|G(x)− G(y)|

)for all x, y ∈ C, where M = M(G,C). (CW1,ω)

We may assume 0 < M <∞, or equivalently that G is nonconstant.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 11 / 35

C1,ω convex extensions of functions

For a mapping G : C ⊂ Rn → Rn we will denote

M(G,C) := supx,y∈C, x 6=y

|G(x)− G(y)|ω(|x− y|)

. (3.1)

We will say that f and G satisfy the property (CW1,ω) on C if either G isconstant, or else 0 < M(G,C) <∞ and

f (x)− f (y)− 〈G(y), x− y〉 ≥ 12|G(x)− G(y)|ω−1

(1

2M|G(x)− G(y)|

)for all x, y ∈ C, where M = M(G,C). (CW1,ω)

We may assume 0 < M <∞, or equivalently that G is nonconstant.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 11 / 35

C1,ω convex extensions of functions

For a mapping G : C ⊂ Rn → Rn we will denote

M(G,C) := supx,y∈C, x 6=y

|G(x)− G(y)|ω(|x− y|)

. (3.1)

We will say that f and G satisfy the property (CW1,ω) on C if either G isconstant, or else 0 < M(G,C) <∞ and

f (x)− f (y)− 〈G(y), x− y〉 ≥ 12|G(x)− G(y)|ω−1

(1

2M|G(x)− G(y)|

)for all x, y ∈ C, where M = M(G,C). (CW1,ω)

We may assume 0 < M <∞, or equivalently that G is nonconstant.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 11 / 35

C1,ω convex extensions of functions

Theorem

Let ω be a modulus of continuity. Let C be a (not necessarily convex)subset of Rn . Let f : C→ R be an arbitrary function, and G : C→ Rn becontinuous, with modulus of continuity ω. Then f has a convex, C1,ω

extension F to all of Rn, with∇F = G on C, if and only if (f ,G) satisfies(CW1,ω) on C.

(CW1,ω)

f (x)− f (y)− 〈G(y), x− y〉 ≥ 12|G(x)− G(y)|ω−1

(1

2M|G(x)− G(y)|

)for all x, y ∈ C,

where

M = M(G,C) := supx,y∈C, x 6=y

|G(x)− G(y)|ω(|x− y|)

.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 12 / 35

C1,ω convex extensions of functions

In particular, for the most important case that ω(t) = t, we have thefollowing.

Corollary

Let C be a (not necessarily convex) subset of Rn. Let f : C→ R be anarbitrary function, and G : C→ Rn be a Lipschitz function. Then f has aconvex, C1,1 extension F to all of Rn, with∇F = G on C, if and only ifthere exists δ > 0 such that

f (x)− f (y)− 〈G(y), x− y〉 ≥ δ |G(x)− G(y)|2, (CW1,1)

for all x, y ∈ C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 13 / 35

C1,ω convex extensions of functions

Our proofs provide good control of the modulus of continuity of thegradients of the extensions F, in terms of that of G. In fact, there exists aconstant k(n) > 0, depending only on the dimension n, such that

M(∇F,Rn) := supx,y∈Rn, x 6=y

|∇F(x)−∇F(y)|ω(|x− y|)

≤ k(n) M(G,C). (3.2)

Recall that

M(G,C) = supx,y∈ C, x 6=y

|G(x)− G(y)|ω(|x− y|)

.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 14 / 35

C1,ω convex extensions of functions

Because convex functions on Rn are not bounded (unless they areconstant), the most usual definitions of norms in the space C1,ω(Rn) are notsuited to estimate convex functions. Therefore, in our setting, for adifferentiable function F : Rn → R, it is natural to define

‖F‖1,ω = |F(0)|+ |∇F(0)|+ supx,y∈Rn, x 6=y

|∇F(x)−∇F(y)|ω(|x− y|)

. (3.3)

With this notation, and assuming 0 ∈ C, our proofs yield

‖F‖1,ω ≤ k(n) (|f (0)|+ |G(0)|+ M(G,C)) . (3.4)

In particular, the norm of the extension F of f that we construct is nearlyoptimal, in the sense that

‖F‖1,ω ≤ k(n) inf{‖ϕ‖1,ω : ϕ ∈ C1,ω(Rn), ϕ|C = f , (∇ϕ)|C = G}. (3.5)

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 15 / 35

C1,ω convex extensions of functions

Because convex functions on Rn are not bounded (unless they areconstant), the most usual definitions of norms in the space C1,ω(Rn) are notsuited to estimate convex functions. Therefore, in our setting, for adifferentiable function F : Rn → R, it is natural to define

‖F‖1,ω = |F(0)|+ |∇F(0)|+ supx,y∈Rn, x 6=y

|∇F(x)−∇F(y)|ω(|x− y|)

. (3.3)

With this notation, and assuming 0 ∈ C, our proofs yield

‖F‖1,ω ≤ k(n) (|f (0)|+ |G(0)|+ M(G,C)) . (3.4)

In particular, the norm of the extension F of f that we construct is nearlyoptimal, in the sense that

‖F‖1,ω ≤ k(n) inf{‖ϕ‖1,ω : ϕ ∈ C1,ω(Rn), ϕ|C = f , (∇ϕ)|C = G}. (3.5)

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 15 / 35

C1 convex extensions of functions

C1 convex extensions of functions

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 16 / 35

C1 convex extensions of functions

ProblemLet C ⊂ Rn be closed, and let f : C→ R, G : C→ Rn be continuousmappings satisfying Whitney’s extension condition (W1), that is,

limx,y∈C, |x−y|→0+

f (x)− f (y)− 〈G(y), x− y〉|x− y|

= 0,

uniformly on compact subsets of C. What additional conditions on f ,G, ifany, will guarantee that there exists a convex function F ∈ C1(Rn) suchthat F = f on C and∇F = G on C?

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 17 / 35

C1 convex extensions of functions

Obstructions: if C is not compact, even in a most nice situation (forinstance, when we assume C to be convex with a smooth boundary, and f isassumed to have C∞ smooth extensions with strictly positive Hessian on aneighborhood of C) there may not be any such F.

There are counterexamples that look like familiar objects:

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 18 / 35

C1 convex extensions of functions

Obstructions: if C is not compact, even in a most nice situation (forinstance, when we assume C to be convex with a smooth boundary, and f isassumed to have C∞ smooth extensions with strictly positive Hessian on aneighborhood of C) there may not be any such F.There are counterexamples that look like familiar objects:

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 18 / 35

C1 convex extensions of functions

The following example is due to Schulz and Schwartz (1979).

Example (Schulz-Schwartz)

Let C = {(x, y) ∈ R2 : x > 0, xy ≥ 1}, and define

f (x, y) = −2√

xy

for every (x, y) ∈ C. The set C is convex and closed, with a nonemptyinterior, and f is convex on a neighborhood of C. However, f does not haveany convex extension to all of R2.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 19 / 35

C1 convex extensions of functions

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 20 / 35

C1 convex extensions of functions

A variation of this bathtub-like example shows that the obstruction persistsif we require that D2f > 0 on a neighborhood of C.

Example

Let C = {(x, y) ∈ R2 : x > 0, xy ≥ 1}, and define

f (x, y) = −2√

xy +1

x + 1+

1y + 1

for every (x, y) ∈ C. The set C is convex and closed, with a nonemptyinterior, and f has a strictly positive Hessian on a neighborhood of C.However, f does not have any convex extension to all of R2.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 21 / 35

C1 convex extensions of functions

So let us assume C compact.The following two geometrical conditions are necessary and sufficient for fto have a convex extension F ∈ C1(Rn) such that F = f on C and ∇F = Gon C.

Namely,

(C) f (x)− f (y) ≥ 〈G(y), x− y〉 for all x, y ∈ C;

(CW1) f (x)− f (y) = 〈G(y), x− y〉 =⇒ G(x) = G(y) for all x, y ∈ C.

TheoremLet C be a compact (not necessarily convex) subset of Rn. Let f : C→ Rbe an arbitrary function, and G : C→ Rn be a continuous mapping. Thenthere exists a convex function F ∈ C1(Rn) with F = f and∇F = G on C ifand only if f and G satisfy the conditions (C) and (CW1) on C.

Moreover, one can arrange that

supx∈Rn|∇F(x)| ≤ k(n) sup

y∈C|G(y)|

(interestingly, this cannot be obtained in general extension problems).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 22 / 35

C1 convex extensions of functions

So let us assume C compact.The following two geometrical conditions are necessary and sufficient for fto have a convex extension F ∈ C1(Rn) such that F = f on C and ∇F = Gon C. Namely,

(C) f (x)− f (y) ≥ 〈G(y), x− y〉 for all x, y ∈ C;

(CW1) f (x)− f (y) = 〈G(y), x− y〉 =⇒ G(x) = G(y) for all x, y ∈ C.

TheoremLet C be a compact (not necessarily convex) subset of Rn. Let f : C→ Rbe an arbitrary function, and G : C→ Rn be a continuous mapping. Thenthere exists a convex function F ∈ C1(Rn) with F = f and∇F = G on C ifand only if f and G satisfy the conditions (C) and (CW1) on C.

Moreover, one can arrange that

supx∈Rn|∇F(x)| ≤ k(n) sup

y∈C|G(y)|

(interestingly, this cannot be obtained in general extension problems).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 22 / 35

C1 convex extensions of functions

So let us assume C compact.The following two geometrical conditions are necessary and sufficient for fto have a convex extension F ∈ C1(Rn) such that F = f on C and ∇F = Gon C. Namely,

(C) f (x)− f (y) ≥ 〈G(y), x− y〉 for all x, y ∈ C;

(CW1) f (x)− f (y) = 〈G(y), x− y〉 =⇒ G(x) = G(y) for all x, y ∈ C.

TheoremLet C be a compact (not necessarily convex) subset of Rn. Let f : C→ Rbe an arbitrary function, and G : C→ Rn be a continuous mapping. Thenthere exists a convex function F ∈ C1(Rn) with F = f and∇F = G on C ifand only if f and G satisfy the conditions (C) and (CW1) on C.

Moreover, one can arrange that

supx∈Rn|∇F(x)| ≤ k(n) sup

y∈C|G(y)|

(interestingly, this cannot be obtained in general extension problems).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 22 / 35

C1 convex extensions of functions

So let us assume C compact.The following two geometrical conditions are necessary and sufficient for fto have a convex extension F ∈ C1(Rn) such that F = f on C and ∇F = Gon C. Namely,

(C) f (x)− f (y) ≥ 〈G(y), x− y〉 for all x, y ∈ C;

(CW1) f (x)− f (y) = 〈G(y), x− y〉 =⇒ G(x) = G(y) for all x, y ∈ C.

TheoremLet C be a compact (not necessarily convex) subset of Rn. Let f : C→ Rbe an arbitrary function, and G : C→ Rn be a continuous mapping. Thenthere exists a convex function F ∈ C1(Rn) with F = f and∇F = G on C ifand only if f and G satisfy the conditions (C) and (CW1) on C.

Moreover, one can arrange that

supx∈Rn|∇F(x)| ≤ k(n) sup

y∈C|G(y)|

(interestingly, this cannot be obtained in general extension problems).Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 22 / 35

C1 convex extensions of functions

Remarks.1. (C) and continuity of G imply (W1).

2. Note that (W1) is a local condition, while (C) and (CW1) are not.Nevertheless, if C is convex with nonempty interior, f is convex, and f ,Gsatisfy (W1) on C, then it can be checked that the conditions (C) and(CW1) are automatically fulfilled. Therefore we have:

CorollaryLet C be a compact convex subset of Rn with non-empty interior. Letf : C→ R be a convex function, and G : C→ Rn be a continuous mappingsatisfying Whitney’s extension condition (W1) on C. Then there exists aconvex function F ∈ C1(Rn) such that F(y) = f (y) and∇F(y) = G(y) forevery y ∈ C.

3. For nonconvex C the nature of the problem is global, in contrast withgeneral extension problems.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 23 / 35

C1 convex extensions of functions

Remarks.1. (C) and continuity of G imply (W1).2. Note that (W1) is a local condition, while (C) and (CW1) are not.

Nevertheless, if C is convex with nonempty interior, f is convex, and f ,Gsatisfy (W1) on C, then it can be checked that the conditions (C) and(CW1) are automatically fulfilled. Therefore we have:

CorollaryLet C be a compact convex subset of Rn with non-empty interior. Letf : C→ R be a convex function, and G : C→ Rn be a continuous mappingsatisfying Whitney’s extension condition (W1) on C. Then there exists aconvex function F ∈ C1(Rn) such that F(y) = f (y) and∇F(y) = G(y) forevery y ∈ C.

3. For nonconvex C the nature of the problem is global, in contrast withgeneral extension problems.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 23 / 35

C1 convex extensions of functions

Remarks.1. (C) and continuity of G imply (W1).2. Note that (W1) is a local condition, while (C) and (CW1) are not.Nevertheless, if C is convex with nonempty interior, f is convex, and f ,Gsatisfy (W1) on C, then it can be checked that the conditions (C) and(CW1) are automatically fulfilled.

Therefore we have:

CorollaryLet C be a compact convex subset of Rn with non-empty interior. Letf : C→ R be a convex function, and G : C→ Rn be a continuous mappingsatisfying Whitney’s extension condition (W1) on C. Then there exists aconvex function F ∈ C1(Rn) such that F(y) = f (y) and∇F(y) = G(y) forevery y ∈ C.

3. For nonconvex C the nature of the problem is global, in contrast withgeneral extension problems.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 23 / 35

C1 convex extensions of functions

Remarks.1. (C) and continuity of G imply (W1).2. Note that (W1) is a local condition, while (C) and (CW1) are not.Nevertheless, if C is convex with nonempty interior, f is convex, and f ,Gsatisfy (W1) on C, then it can be checked that the conditions (C) and(CW1) are automatically fulfilled. Therefore we have:

CorollaryLet C be a compact convex subset of Rn with non-empty interior. Letf : C→ R be a convex function, and G : C→ Rn be a continuous mappingsatisfying Whitney’s extension condition (W1) on C. Then there exists aconvex function F ∈ C1(Rn) such that F(y) = f (y) and∇F(y) = G(y) forevery y ∈ C.

3. For nonconvex C the nature of the problem is global, in contrast withgeneral extension problems.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 23 / 35

C1 convex extensions of functions

Remarks.1. (C) and continuity of G imply (W1).2. Note that (W1) is a local condition, while (C) and (CW1) are not.Nevertheless, if C is convex with nonempty interior, f is convex, and f ,Gsatisfy (W1) on C, then it can be checked that the conditions (C) and(CW1) are automatically fulfilled. Therefore we have:

CorollaryLet C be a compact convex subset of Rn with non-empty interior. Letf : C→ R be a convex function, and G : C→ Rn be a continuous mappingsatisfying Whitney’s extension condition (W1) on C. Then there exists aconvex function F ∈ C1(Rn) such that F(y) = f (y) and∇F(y) = G(y) forevery y ∈ C.

3. For nonconvex C the nature of the problem is global, in contrast withgeneral extension problems.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 23 / 35

A geometrical application

A geometrical application

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 24 / 35

A geometrical application

We can use the above results to solve a geometrical problem concerningcharacterizations of subsets K of Rn which can be interpolated byboundaries of C1 (resp. C1,1) convex bodies, with prescribed unit outernormals, in the spirit of the mentioned result of Ghomi’s for K = asubmanifold.

Namely, if C is a subset of Rn and we are given a Lipschitz mapN : C→ Rn such that |N(y)| = 1 for every y ∈ C, what conditions on Cand N are necessary and sufficient for C to be contained in the boundary ofa C1,1 convex body V such that 0 ∈ int(V) and N(y) is outwardly normal to∂V at y for every y ∈ C?

A suitable set of conditions is:

(O) 〈N(y), y〉 ≥ δ for all y ∈ C;

(KW1,1) 〈N(y), y− x〉 ≥ δ|N(y)− N(x)|2 for all x, y ∈ C,

for some δ > 0.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 25 / 35

A geometrical application

We can use the above results to solve a geometrical problem concerningcharacterizations of subsets K of Rn which can be interpolated byboundaries of C1 (resp. C1,1) convex bodies, with prescribed unit outernormals, in the spirit of the mentioned result of Ghomi’s for K = asubmanifold.

Namely, if C is a subset of Rn and we are given a Lipschitz mapN : C→ Rn such that |N(y)| = 1 for every y ∈ C, what conditions on Cand N are necessary and sufficient for C to be contained in the boundary ofa C1,1 convex body V such that 0 ∈ int(V) and N(y) is outwardly normal to∂V at y for every y ∈ C?

A suitable set of conditions is:

(O) 〈N(y), y〉 ≥ δ for all y ∈ C;

(KW1,1) 〈N(y), y− x〉 ≥ δ|N(y)− N(x)|2 for all x, y ∈ C,

for some δ > 0.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 25 / 35

A geometrical application

We can use the above results to solve a geometrical problem concerningcharacterizations of subsets K of Rn which can be interpolated byboundaries of C1 (resp. C1,1) convex bodies, with prescribed unit outernormals, in the spirit of the mentioned result of Ghomi’s for K = asubmanifold.

Namely, if C is a subset of Rn and we are given a Lipschitz mapN : C→ Rn such that |N(y)| = 1 for every y ∈ C, what conditions on Cand N are necessary and sufficient for C to be contained in the boundary ofa C1,1 convex body V such that 0 ∈ int(V) and N(y) is outwardly normal to∂V at y for every y ∈ C?

A suitable set of conditions is:

(O) 〈N(y), y〉 ≥ δ for all y ∈ C;

(KW1,1) 〈N(y), y− x〉 ≥ δ|N(y)− N(x)|2 for all x, y ∈ C,

for some δ > 0.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 25 / 35

Ideas of the proofs

Ideas of the proofs

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 26 / 35

Ideas of the proofs

Necessity, C1 case: If f ∈ C1(Rn) is convex then

f (x)− f (y) = 〈f (y), x− y〉 =⇒ ∇f (x) = ∇f (y).

Indeed, up to adding an affine function we may assume f (x) = f (y) = 0,∇f (y) = 0. Then, by convexity and differentiability,∇f (x) = 0.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 27 / 35

Ideas of the proofs

Necessity, C1,ω case: Let f ∈ C1,ω(Rn) be convex. Then

f (x)−f (y)−〈∇f (y), x−y〉 ≥ 12|∇f (x)−∇f (y)|ω−1

(1

2M|∇f (x)−∇f (y)|

)for all x, y ∈ Rn, where

M = M(∇f ,Rn) = supx,y∈Rn, x 6=y

|∇f (x)−∇f (y)|ω (|x− y|)

.

Proof: ... a quantitative refinement of the preceding argument.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 28 / 35

Ideas of the proofs

Sufficiency: both the C1 and the C1,ω results use the following strategy:

1. Since (C) + (CW1) =⇒ (W1), and(CW1,ω)+ ω-continuity of G =⇒ (W1,ω), by using Whitney’s extensiontheorems we may assume f ∈ C1(Rn) (resp. f ∈ C1,ω(Rn) satisfies (C) and(CW1) on C (resp. (CW1,ω) on C).

2. Define m(f ) : Rn → R by

m(f )(x) = supy∈C{f (y) + 〈∇f (y), x− y〉}.

This function is finite everywhere (use (CW1,ω) if necessary), and convex,and we have m(f ) = f on C. (In the case that C is a convex body m(f ) isthe minimal convex extension of f to Rn.)

If the function m(f ) were differentiable on Rn, there would be nothing elseto say. Unfortunately, there are examples showing that m(f ) need not bedifferentiable outside C, even when C is convex and f satisfies (CW1).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 29 / 35

Ideas of the proofs

Sufficiency: both the C1 and the C1,ω results use the following strategy:

1. Since (C) + (CW1) =⇒ (W1), and(CW1,ω)+ ω-continuity of G =⇒ (W1,ω), by using Whitney’s extensiontheorems we may assume f ∈ C1(Rn) (resp. f ∈ C1,ω(Rn) satisfies (C) and(CW1) on C (resp. (CW1,ω) on C).

2. Define m(f ) : Rn → R by

m(f )(x) = supy∈C{f (y) + 〈∇f (y), x− y〉}.

This function is finite everywhere (use (CW1,ω) if necessary), and convex,and we have m(f ) = f on C. (In the case that C is a convex body m(f ) isthe minimal convex extension of f to Rn.)

If the function m(f ) were differentiable on Rn, there would be nothing elseto say. Unfortunately, there are examples showing that m(f ) need not bedifferentiable outside C, even when C is convex and f satisfies (CW1).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 29 / 35

Ideas of the proofs

Sufficiency: both the C1 and the C1,ω results use the following strategy:

1. Since (C) + (CW1) =⇒ (W1), and(CW1,ω)+ ω-continuity of G =⇒ (W1,ω), by using Whitney’s extensiontheorems we may assume f ∈ C1(Rn) (resp. f ∈ C1,ω(Rn) satisfies (C) and(CW1) on C (resp. (CW1,ω) on C).

2. Define m(f ) : Rn → R by

m(f )(x) = supy∈C{f (y) + 〈∇f (y), x− y〉}.

This function is finite everywhere (use (CW1,ω) if necessary), and convex,and we have m(f ) = f on C. (In the case that C is a convex body m(f ) isthe minimal convex extension of f to Rn.)

If the function m(f ) were differentiable on Rn, there would be nothing elseto say. Unfortunately, there are examples showing that m(f ) need not bedifferentiable outside C, even when C is convex and f satisfies (CW1).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 29 / 35

Ideas of the proofs

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 30 / 35

Ideas of the proofs

Sufficiency: both the C1 and the C1,ω results use the following strategy:1. We may assume f ∈ C1(Rn) satisfies (C) and (CW1) on C (resp.(CW1,ω) on C).

2. Define m(f ) : Rn → R by

m(f )(x) = supy∈C{f (y) + 〈∇f (y), x− y〉}.

This function is and convex on Rn, and we have m(f ) = f on C.

3. Show that m(f ) is differentiable on C (resp. ω-differentiable on C; infact:

m(f )(x)− m(f )(x0)− 〈∇f (x0), x− x0〉 ≤ 4Mω(|x− x0|)|x− x0|

for every x0 ∈ C, x ∈ Rn).

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 31 / 35

Ideas of the proofs

4. Assume for the time being that m(f ) is coercive, i.e.,

lim|x|→∞

m(f )(x) =∞.

5. We want to construct a differentiable function ϕ (resp. ϕ ∈ C1,ω(Rn))

such that ϕ = 0 and∇ϕ = 0 on C, and f + ϕ ≥ m(f ) on Rn, and thenapply the following:

Theorem (Kirchheim-Kristensen)

If H : Rn → R is a differentiable function on Rn such thatlim|x|→+∞H(x) = +∞, then the convex envelope conv(H) of H is aconvex function of class C1(Rn). If we further assume H to beω-differentiable then conv(H) ∈ C1,ω(Rn).

Then by taking F = conv(f + ϕ) we would obtain a convex extension of fof class C1 (resp. C1,ω), such that∇F = G on C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 32 / 35

Ideas of the proofs

4. Assume for the time being that m(f ) is coercive, i.e.,

lim|x|→∞

m(f )(x) =∞.

5. We want to construct a differentiable function ϕ (resp. ϕ ∈ C1,ω(Rn))

such that ϕ = 0 and∇ϕ = 0 on C, and f + ϕ ≥ m(f ) on Rn, and thenapply the following:

Theorem (Kirchheim-Kristensen)

If H : Rn → R is a differentiable function on Rn such thatlim|x|→+∞H(x) = +∞, then the convex envelope conv(H) of H is aconvex function of class C1(Rn). If we further assume H to beω-differentiable then conv(H) ∈ C1,ω(Rn).

Then by taking F = conv(f + ϕ) we would obtain a convex extension of fof class C1 (resp. C1,ω), such that∇F = G on C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 32 / 35

Ideas of the proofs

4. Assume for the time being that m(f ) is coercive, i.e.,

lim|x|→∞

m(f )(x) =∞.

5. We want to construct a differentiable function ϕ (resp. ϕ ∈ C1,ω(Rn))

such that ϕ = 0 and∇ϕ = 0 on C, and f + ϕ ≥ m(f ) on Rn, and thenapply the following:

Theorem (Kirchheim-Kristensen)

If H : Rn → R is a differentiable function on Rn such thatlim|x|→+∞H(x) = +∞, then the convex envelope conv(H) of H is aconvex function of class C1(Rn). If we further assume H to beω-differentiable then conv(H) ∈ C1,ω(Rn).

Then by taking F = conv(f + ϕ) we would obtain a convex extension of fof class C1 (resp. C1,ω), such that∇F = G on C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 32 / 35

Ideas of the proofs

4. Assume for the time being that m(f ) is coercive, i.e.,

lim|x|→∞

m(f )(x) =∞.

5. We want to construct a differentiable function ϕ (resp. ϕ ∈ C1,ω(Rn))

such that ϕ = 0 and∇ϕ = 0 on C, and f + ϕ ≥ m(f ) on Rn, and thenapply the following:

Theorem (Kirchheim-Kristensen)

If H : Rn → R is a differentiable function on Rn such thatlim|x|→+∞H(x) = +∞, then the convex envelope conv(H) of H is aconvex function of class C1(Rn). If we further assume H to beω-differentiable then conv(H) ∈ C1,ω(Rn).

Then by taking F = conv(f + ϕ) we would obtain a convex extension of fof class C1 (resp. C1,ω), such that∇F = G on C.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 32 / 35

Ideas of the proofs

6. How can we construct a differentiable function ϕ (resp. a functionϕ ∈ C1,ω(Rn)) such that ϕ = 0 and ∇ϕ = 0 on C and f + ϕ ≥ m(f ) onRn?

In the C1 case, apply Whitney’s approximation theorem and make someeasy calculations.

In the C1,ω case, we may assume C closed. Take a decomposition of Rn \Cinto Whitney cubes {Qj,Q∗j }j and the asociated Whitney partition of unity{ϕj}j. Define

pj := supx∈Q∗

j

|f (x)− m(f )(x)|.

Then one can show that the function

ϕ(x) :=

{ ∑j pj ϕj(x) if x ∈ Rn \ C,

0 if x ∈ C

is of class C1,ω(Rn) with ϕ = 0 and∇ϕ = 0 on C, and

ϕ+ f ≥ m(f ) on Rn.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 33 / 35

Ideas of the proofs

6. How can we construct a differentiable function ϕ (resp. a functionϕ ∈ C1,ω(Rn)) such that ϕ = 0 and ∇ϕ = 0 on C and f + ϕ ≥ m(f ) onRn?

In the C1 case, apply Whitney’s approximation theorem and make someeasy calculations.

In the C1,ω case, we may assume C closed. Take a decomposition of Rn \Cinto Whitney cubes {Qj,Q∗j }j and the asociated Whitney partition of unity{ϕj}j. Define

pj := supx∈Q∗

j

|f (x)− m(f )(x)|.

Then one can show that the function

ϕ(x) :=

{ ∑j pj ϕj(x) if x ∈ Rn \ C,

0 if x ∈ C

is of class C1,ω(Rn) with ϕ = 0 and∇ϕ = 0 on C, and

ϕ+ f ≥ m(f ) on Rn.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 33 / 35

Ideas of the proofs

6. How can we construct a differentiable function ϕ (resp. a functionϕ ∈ C1,ω(Rn)) such that ϕ = 0 and ∇ϕ = 0 on C and f + ϕ ≥ m(f ) onRn?

In the C1 case, apply Whitney’s approximation theorem and make someeasy calculations.

In the C1,ω case, we may assume C closed. Take a decomposition of Rn \Cinto Whitney cubes {Qj,Q∗j }j and the asociated Whitney partition of unity{ϕj}j. Define

pj := supx∈Q∗

j

|f (x)− m(f )(x)|.

Then one can show that the function

ϕ(x) :=

{ ∑j pj ϕj(x) if x ∈ Rn \ C,

0 if x ∈ C

is of class C1,ω(Rn) with ϕ = 0 and ∇ϕ = 0 on C, and

ϕ+ f ≥ m(f ) on Rn.

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 33 / 35

Ideas of the proofs

7. General case: m(f ) is not necessarily coercive. Use the following:

PropositionLet g : Rn → R be a convex function, and assume g is not affine. Thenthere exist a linear function ` : Rn → R, a positive integer k ≤ n, a linearsubspace X ⊆ Rn of dimension k, and a convex function c : X → R suchthat

lim|x|→∞

c(x) =∞, and g = `+ c ◦ P,

where P : Rn → X is the orthogonal projection.

So, by using the projection P, we may write

m(f ) = `+ c ◦ P,

then we may further assume that ` = 0, and essentially the above proof,with c in place of m(f ), gives a C1,ω (resp. C1) convex extension Ψ ofc|P(C)

. Define F = Ψ ◦ P and check that F extends f .

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 34 / 35

Ideas of the proofs

7. General case: m(f ) is not necessarily coercive. Use the following:

PropositionLet g : Rn → R be a convex function, and assume g is not affine. Thenthere exist a linear function ` : Rn → R, a positive integer k ≤ n, a linearsubspace X ⊆ Rn of dimension k, and a convex function c : X → R suchthat

lim|x|→∞

c(x) =∞, and g = `+ c ◦ P,

where P : Rn → X is the orthogonal projection.

So, by using the projection P, we may write

m(f ) = `+ c ◦ P,

then we may further assume that ` = 0, and essentially the above proof,with c in place of m(f ), gives a C1,ω (resp. C1) convex extension Ψ ofc|P(C)

. Define F = Ψ ◦ P and check that F extends f .

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 34 / 35

Ideas of the proofs

Thank you for your attention!

Daniel Azagra and Carlos Mudarra C1,ω convex extensions of functions Haifa, May 30, 2015 35 / 35