HESSIAN MEASURES OF CONVEX FUNCTIONS AND APPLICATIONS TO AREA MEASURES

DOI: 10.1112/S0000000000000000

HESSIAN MEASURES OF CONVEX FUNCTIONS AND APPLICATIONS TO AREA MEASURES

ANDREA COLESANTIANDDANIEL HUG

ABSTRACT

The Hessian measures of a (semi-)convex function can be introduced as coefficients of a local Steiner formula.

We continue the investigation of Hessian measures by providing a geometric characterization of the support of these measures. Then we explore the Radon-Nikodym derivative and the absolute continuity of Hessian measures with respect to Lebesgue measure. As special cases of our results, we recover known results for surface area measures of convex bodies.

1. Introduction

A natural way to introduce the Hessian measures of a convex function is through a local Steiner formula. Letube a convex function defined in an open convex subsetΩof the Euclidean spaceR^d(d≥2). For each pointxinΩ, we denote by∂u(x)the subdifferential ofuatx. Ifηis a Borel subset ofΩandρ >0, then the set

Pρ(u, η) ={x+ρζ : x∈η , ζ∈∂u(x)}

is measurable and its measure can be expressed as a polynomial of degreedinρ, that is H^d(Pρ(u, η)) =

d

X

k=0

d k

Fk(u, η)ρ^d−k, (1.1) whereH^ddenotesd-dimensional Hausdorff measure (Lebesgue measure). The coefficients F0(u,·), . . . , Fd(u,·)are nonnegative Borel measures, which are called the Hessian mea- sures ofu. When u∈ C²(Ω), the Hessian measures are simply the integrals of the ele- mentary symmetric functions of the eigenvalues of the second order differentialD²uofu;

more precisely, d

k

F_k(u, η) = Z

η

σ_d−k(D²u(x))H^d(dx), k= 0, . . . , d , (1.2) where

σ_j(D²u(x)) = X

1≤i1<···<ij≤d

λ_i1· · ·λ_ij, j= 1, . . . , d ,

andλ1, . . . , λdare the eigenvalues of the Hessian matrixD²u(x)(we also putσ0= 1).

Though in this paper we are concerned mainly with those aspects of Hessian measures which are related to the theory of convex bodies, let us recall that another interesting fea- ture of these measures is their connection with fully non-linear elliptic partial differential

2000 Mathematics Subject Classification 52A20 (primary), 26B25, 28A78 (secondary).

ANDREA COLESANTI AND DANIEL HUG

equations. For this topic we refer the reader to [18], [8] and the bibliographical references in these papers.

Formula (1.1) emphasizes the analogy between the Hessian measures of a convex func- tion and the curvature and (surface) area measures of a convex body (see [15]). Indeed, curvature and area measures also admit a characterization as coefficients of a local Steiner formula. Another link is given by a pair of formulas proved in [7] that we recall here. For a convex bodyK (nonempty compact convex set inR^d), let us denote byCj(K,·)and Sj(K,·),j= 0, . . . , d−1, the curvature and area measures ofK, respectively. Let bd(K) denote the boundary ofK. IfdKis the distance function ofK, then

C_j(K, η) =dF_j(d_K, η∩bd(K)), j= 0, . . . , d−1, (1.3) for every Borel setη⊂R^d; moreover, ifhK is the support function ofK, then

Sj(K, ω) =dFd−j(hK,ω)ˆ , j= 0, . . . , d−1, (1.4) whereω is a Borel subset of the unit sphereS^d−1andωˆ := {tν : t ∈ [0,1], ν ∈ ω}.

In some cases such relations allow one to deduce results regarding curvature and area measures from general properties of Hessian measures; examples are given in [7] and in the present paper.

Our first result is a geometric description of the support of the Hessian measures of convex functions. Such a result parallels corresponding characterizations of the support of curvature and area measures of convex bodies (see, for instance, Theorems 4.6.1 and 4.6.3 in [15]). For a convex function u, defined in an open convex set Ωand for j ∈ {0, . . . , d−1}, we introduce the set of itsj-extreme points. A pointx ∈ Ωis calledj- extreme if there exists no(j+ 1)-dimensional ball centred atxand contained inΩ, on whichuis affine. Thus, extreme points of convex functions can be seen in analogy to extreme boundary points of convex bodies. Our result, stated as Theorem 1, claims that the support of the Hessian measureF_j(u,·)is the closure, inΩ, of the set ofj-extreme points of u. The proof of this fact is achieved by using a representation of Hessian measures, established in [6], and by an inspection of the proof for the characterization of the support of curvature measures of convex bodies.

The investigation of Hessian measures thus leads to a unified view on the characteri- zation of the support of curvature and surface area measures of convex bodies. The latter results were first obtained in [13], [14]; see also [17].

In the second and main part of the paper, we investigate the Radon-Nikodym derivative and the absolute continuity of Hessian measures of (semi-)convex functions with respect tod-dimensional Hausdorff measureH^dinR^d. First, we describe explicitly the absolutely continuous part (the Radon-Nikodym derivative) of the Hessian measures with respect to H^d, in terms of pointwise second derivatives of the involved function. More precisely, consider a (semi-)convex functionudefined in a convex open subset of R^d. It is well- known thatuis second order differentiable in Aleksandrov’s sense at almost every point ofΩ(see [2], [3], [4], [9]); hence, forH^d-almost everyxinΩ, the Hessian matrixD²u(x) is defined. In Theorem 2 we show that, forj∈ {0, . . . , d}andH^d-almost everywhere, the Radon-Nikodym derivativedF_d−j(u,·)/dH^dofF_d−j(u,·)with respect tod-dimensional Hausdorff measure satisfies

d j

dF_d−j

dH^d =σj(D²u). (1.5)

This result has a natural counterpart for curvature and area measures of convex bodies; see

HESSIAN MEASURES OF CONVEX FUNCTIONS

Theorems 3.2 and 3.5 in [11]. In fact, part of the information contained in Theorem 3.5 in [11] can be deduced from (1.5) via (1.4). An explicit description of the singular part of the Hessian measures with respect toH^dis implicitly contained in the proof of Theorem 2.

Our next result was inspired by a theorem which has been proved by Weil in [19] for area measures, and which has recently been established for curvature measures in [12].

Roughly speaking, for curvature measures the result states that the absolute continuity of the mean curvature measure of a convex set and some integrability assumption for the mean curvature (in Aleksandrov’s sense) together imply the absolute continuity of further (lower order) curvature measures of the given set. In the case of Hessian measures, the result can be described as follows. Assume that the Hessian measureFd−1(u,·)of a (semi-)convex functionu, defined inΩ, is absolutely continuous with respect toH^d and that, for some p≥2,

dF_d−1

dH^d ∈L^p_loc(Ω). (1.6)

By Theorem 2, the Radon-Nikodym derivative in (1.6) coincides with the Laplacian of u(up to the constant factord),H^d-almost everywhere. Theorem 3 states that for every j≤d−1such thatj≥d−p,F_j(u,·)is absolutely continuous with respect toH^dand

dFj

dH^d ∈L^q_loc(Ω),

for everyq ≥ 1such that(d−j)q ≤p. It is clear from the examples given in [12] that this result (in a sense) is best possible. As a consequence, we obtain Weil’s theorem for area measures by applying our result in the special case of Hessian measures of support functions. Curiously, it does not seem to be possible to derive the corresponding result for curvature measures in a similar way.

2. The support of Hessian measures

In the following, we work in Euclidean spaces with scalar producth·,·iand with norm k · k. A closed Euclidean ball inR^d with centrexand radiusr ≥ 0 will be denoted by B^d(x, r). We writeS^d−1for the Euclidean unit sphere. The support of a Borel measureµ will be denoted by suppµ. It is defined as the complement of the largest open set on which the measure vanishes. LetK^d_odenote the set of convex bodies with nonempty interiors. The geometric characterization of the support of thejth curvature measure of a convex body K ∈ K^d_o involves the closure of the setext_j(K)ofj-extreme boundary points ofK. A pointx ∈ bd(K)is called a j-extreme boundary point ofK ifxis not the centre of a (j+ 1)-dimensional ball contained in K; see [15, p. 65] for equivalent definitions. The following result is due to R. Schneider [14].

THEOREM(Schneider). LetK∈ K^d_oandj∈ {0, . . . , d−1}. Then

suppCj(K,·) = cl(extj(K)). (2.1) We now introduce the appropriate notion which is needed for the description of the support of a Hessian measure. LetΩ⊂R^dbe open and convex, and letu: Ω →Rbe a convex function. Then we say thatx∈Ωis aj-extreme point ofuif there is no(j+ 1)- dimensional ball centred atxand contained inΩon whichuis an affine function. The set of allj-extreme points ofuis denoted by extj(u). For a setA⊂Ω, we write clΩ(A)for the closure ofAwith respect toΩ, that is clΩ(A) = Ω∩cl(A). All other definitions and notations are as in [15], [6], [7].

ANDREA COLESANTI AND DANIEL HUG

THEOREM1. LetΩ⊂R^d be open and convex, letu: Ω→Rbe a convex function, and letj∈ {0, . . . , d−1}. Then

suppF_j(u,·) = cl_Ω(ext_j(u)). (2.2) Proof. The generalized Hessian measureΘ_j(u,·)of uis introduced in [6, Theorem 3.1] as coefficient measure of a Steiner formula. It is a measure on the Borel subsets of Ω×R^dwhich satisfiesFj(u,·) = Θj(u,· ×R^d). Therefore [6, Theorem 3.1] especially yields an integral representation forFj(u,·)(cf. (3.2)). A similar integral representation is available for the curvature measure Cj(K,·)of a closed convex set K ⊂ R^d+1; see Theorem 3.2 in [11].

Now we compare the integral representations forFj(u,·)and forCj(epi(u),·), where epi(u) :={(x, z)∈Ω×R:z ≥u(x)}is the epigraph ofu. Up to the homeomorphism gu : Ω→ graph(u) ⊂R^d+1,x7→ (x, u(x)), these two representations differ only by a positive and finite function under the integral. Therefore, we obtain

suppFj(u,·) =suppCj(epi(u), gu(·)). (2.3) An inspection of the proof of Theorem 4.6.1 in [15] shows that all arguments involved in that proof are of a local nature, and hence

suppCj(epi(u), gu(·)) =g_u⁻¹(cl(extj(epi(u)))). (2.4) Since obviously

cl(ext_j(epi(u))) =g_u(cl_Ω(ext_j(u))) , (2.5) the proof follows by combining (2.3) – (2.5).

In [14] and [15], the proof of (2.1) is based on a combination of geometric arguments and on an integral-geometric mean value formula (Crofton formula). Instead of trying to repeat this argument by using a Crofton formula for convex functions (see [7, Theorem 2.1]), we applied the relationship between the Hessian measures of a convex function and the curvature measures of the epigraph of this function. Apart from providing a new result for Hessian measures, a major advantage of the present point of view is that we now obtain the characterization of the support of the surface area measures of convex bodies quite easily. Note that despite certain analogies, the proof of Theorem 4.6.3 in [15] for surface area measures (see also [13]) is slightly more involved than the proof of Theorem 4.6.1 in [15] concerning curvature measures.

To describe the support of the area measures of a convex bodyK ∈ K^d_o, we recall that a unit vectorν ∈S^d−1is aj-extreme normal vector ofKif and only if there do not exist j+ 2linearly independent normal vectorsν1, . . . , νj+2at one and the same boundary point ofKsuch thatν=ν1+. . .+νj+2. An equivalent condition involving the support function h(K,·)ofKis thatν =ν1+. . .+νj+2 with linearly independent vectorsν1, . . . , νj+2

implies thath(K, ν)< h(K, ν1) +. . .+h(K, νj+2); cf. [16, p. 278] or [15, Section 2.2].

Letextn_j(K)denote the set ofj-extreme normal vectors ofK∈ K^d_o.

In view of another application in Section 3, the following lemma is stated in greater generality than needed for the proof of (2.9).

LEMMA1. LetK∈ K_o^dandj∈ {1, . . . , d}. Then dF_j(h_K,·) =j

Z∞ 0

Z

S^d−1

1{rν∈ ·}r^j−1S_d−j(K, dν)dr .

HESSIAN MEASURES OF CONVEX FUNCTIONS

Proof. LetR : (0,∞)×S^d−1 → R^d be defined byR(r, ν) := rν. Then, as in the proof of Corollary 5.10 in [7], one obtains

dF_j(h_K, R((0,1]×ω)) =S_d−j(K, ω) (2.6) for any Borel setω⊂S^d−1. Since∂h_K(tν) =∂h_K(ν)fort >0andν ∈S^d−1, we find that

Pρ(hK, R((0, b]×ω)) =bPρ/b(hK, R((0,1]×ω))

for anyρ, b > 0 and any Borel setω ⊂ S^d−1. Thus, applying the Steiner formula for Hessian measures and comparing coefficients, we get

Fj(hK, R((0, b]×ω)) =b^jFj(hK, R((0,1]×ω)). (2.7) From (2.6) and (2.7) we deduce that

dFj(hK, R(γ)) =j Z_∞

0

Z

S^d−1

1{(r, ν)∈γ}r^j−1Sd−j(K, dν)dr (2.8) withγ= (0, b]×ω. Since on both sides of (2.8), we have measures with respect toγ, the assertion follows by general measure theoretic extension arguments.

THEOREM(Schneider). LetK∈ K^d_oandj∈ {0, . . . , d−1}. Then

suppS_d−1−j(K,·) = cl(extnj(K)). (2.9) Proof. Forω ⊂S^d−1, we putω˜ :={tν : t∈ (0,1), ν ∈ω}. Lemma 1 implies that ν /∈ suppS_d−1−j(K,·)if and only if there is an open setω ⊂ S^d−1 withν ∈ ωsuch thatF_j+1(h_K,ω) = 0. By Theorem 1, this is equivalent to the following condition: for˜ any ζ ∈ ω, there exist vectors˜ ν1, . . . , νj+1 ∈ R^d and a number > 0, all depending onζ, such thatζ, ν1, . . . , νj+1are linearly independent andhK is an affine function on ζ+B^d(0, )∩lin{ν1, . . . , νj+1}. This is the same as requiring, for anyζ∈ω, the existence˜ of vectors ζ1, . . . , ζj+1 ∈ R^d and of a number δ > 0, all depending on ζ, such that (ζ, ζ1, . . . , ζj+1)is an orthonormal system andh(K,·)is a linear function on the convex cone

N :=

λ ζ+B^d(0, δ)∩lin{ζ₁, . . . ζ_j+1}

:λ≥0 .

But for any ζ ∈ ω, this is equivalent to the existence of linearly independent vectors˜ w1, . . . , wj+2such thatζ/kζk=w1+. . .+wj+2and

h(K, ζ/kζk) =h(K, w₁) +. . .+h(K, w_j+2), that isζ/kζk∈/ extn_j(K).

3. Radon-Nikodym derivative and absolute continuity

Letµbe a nonnegative Borel measure onΩ⊂ R^d. Ifµis absolutely continuous with respect toH^d, then we writeµ H^dand denote bydµ/dH^dthe density ofµwith respect toH^d. In general,µcan be written as the sum of two measuresµ^a, µ^s, whereµ^a H^d andµ^a, µ^s are mutually singular. The density dµ^a/dH^d is called the Radon-Nikodym derivative ofµwith respect toH^d; see [9] for details.

In the following theorem, we determine the absolutely continuous part of the Hessian measures of convex functions with respect toH^d. The singular part is explicitly described by equation (3.4) in the subsequent proof.

ANDREA COLESANTI AND DANIEL HUG

THEOREM 2. LetΩ ⊂ R^d be an open convex set, and let u : Ω → Rbe a convex function. Then, forj ∈ {0, . . . , d} the Radon-Nikodym derivative of ^d_j

F_d−j(u,·)with respect to H^d is given by σj(D²u), where D²u(x)is the Hessian matrix of u atxin Aleksandrov’s sense, forH^d-almost everyx∈Ω.

Proof. In the following, we adopt the notation from [6], except that we writedfor the dimension instead of n. So Nor(u) ⊂ R^d+1 ×R^d+1 is the normal bundle of u, K₁(X, V), . . . , K_d(X, V) are the generalized curvatures onNor(u), which are defined forH^d-almost every (X, V) ∈ Nor(u),U₁, . . . , U_d denote the corresponding eigenvec- tors (depending on(X, V)), and E₁, . . . , E_d+1 is the standard basis ofR^d+1. The span ofE₁, . . . , E_dis identified withR^d. As in [6],U₁, . . . , U_d are chosen in such a way that (U1, . . . , Ud, V)is an orthonormal basis which is negatively oriented with respect to the standard basis.

The generalized graph ofuis

Γ(u) ={(x, p)∈Ω×R^d:p∈∂u(x)},

which is homeomorphic toNor(u)via the locally bilipschitz mapT : Γ(u) → Nor(u) given by

T(x, p) =

(x, u(x)),(1 +kpk²)^−1/2(p,−1)

(3.1) with inverse

T˜(X, V) = X− hX, Ed+1iEd+1, Ed+1− hV, Ed+1i⁻¹V .

The casej = 0of Theorem 4.1 is covered by Corollary 3.2 in [6]. Therefore we suppose thatj∈ {1, . . . , d}in the following. By a special case of Theorem 3.1 in [6],

d j

F_d−j(u,·) = Z

Nor(u)

1{X− hX, Ed+1iEd+1∈ ·}

− 1

hV, Ed+1i j

× X

1≤i1<···<ij≤d

K_i1(X, V)· · ·K_ij(X, V) Qd

i=1

p1 +Ki(X, V)²

×Di₁...i_j(X, V)H^d(d(X, V)). (3.2) For a definition ofD_i1...ij, we refer to [6, p. 3246–7]. We now splitNor(u)into the mea- surable sets

Nor^a(u) :={(X, V)∈Nor(u) :Ki(X, V)<∞fori= 1, . . . , d}

and

Nor^s(u) := Nor(u)\Nor^a(u) ;

cf. [11]. LetΠ1:R^d+1×R^d+1→R^d+1be defined byΠ1(X, Y) :=X. The approximate Jacobian of the mapΠ1◦T˜ : Nor(u)→Ω,(X, V)7→X− hX, Ed+1iEd+1satisfies

apJd(Π1◦T˜)(X, V) =

d

^

i=1

1

p1 +Ki(X, V)²Ai

= (−hV, Ed+1i)

d

Y

i=1

1

p1 +K_i(X, V)²,

HESSIAN MEASURES OF CONVEX FUNCTIONS

whereA_i:=U_i− hU_i, E_d+1iE_d+1is defined as in [6]. Next we assert that d

j

F_d−j^a (u,·) = Z

Nor^a(u)

1{Π1◦T˜(X, V)∈ ·}apJd(Π1◦T)(X, V˜ )

×

− 1

hV, Ed+1i j+1

X

1≤i1<···<ij≤d

K_i1(X, V)· · ·K_ij(X, V)

×Di₁...i_j(X, V)H^d(d(X, V)) (3.3) and

d j

F_d−j^s (u,·) = Z

Nor^s(u)

1{Π1◦T˜(X, V)∈ ·}

− 1

hV, Ed+1i ^j

× X

1≤i1<···<ij≤d

K_i1(X, V)· · ·K_ij(X, V) Qd

i=1

p1 +K_i(X, V)²

×Di₁...i_j(X, V)H^d(d(X, V)). (3.4) To verify the equations (3.3) and (3.4), we argue similarly as in [11]. LetD²(u)denote the set of all pointsx∈Ωat whichuis second order differentiable in the Aleksandrov sense.

For(X, V) := T(x,∇u(x))andx ∈ D²(u), we getKj(X, V) <∞forj = 1, . . . , d, since thenXis a normal boundary point of the epigraph ofu. Therefore,F_d−j^s (u,·)van- ishes onD²(u). Moreover, sinceH^d(Ω\ D²(u)) = 0, the coarea formula yields that

d j

F_d−j^a (u,·) = Z

Ω

1{x∈ ·}d_j(x)H^d(dx), where

dj(x) = (−hV, Ed+1i)^−(j+1) X

1≤i1<···<ij≤d

Ki₁(X, V)· · ·Ki_j(X, V)Di₁...i_j(X, V) forx ∈ D²(u)and(X, V) = T(x,∇u(x)). ThusF_d−j^a (u,·)is concentrated onD²(u).

This concludes the proof of (3.3) and (3.4).

It remains to show that

d_j(x) =σ_j(D²u(x)) (3.5)

forH^d-almost everyx∈ D²(u). The proof of (3.5) is split into two main steps.

Step 1. Letx∈ D²(u)and(X, V) =T(x,∇u(x))be fixed. As in [6], we can define Bj:= Kj(X, V)

−hV, E_d+1iBj:= Kj(X, V)

−hV, E_d+1i

Uj−hUj, Ed+1i hV, E_d+1iV

,

whereKj(X, V) <∞is used. SinceA1, . . . , Adis a basis ofR^d, there are coefficients βij,i, j∈ {1, . . . , d}, such that

Bj=

d

X

i=1

βijAi, j= 1, . . . , d . (3.6) For1≤i1<· · ·< ij≤dwe putI:= (i1, . . . , ij)and

sgn(I) := sgn(i1. . . ijij+1. . . id),

where1 ≤ ij+1 < · · · < id ≤ d is complementary to I with respect to {1, . . . , d}.

ANDREA COLESANTI AND DANIEL HUG

Omitting the argument(X, V), we thus obtain

− 1

hV, Ed+1i j

Ki1· · ·KijDi1...ijE1∧ · · · ∧Ed

= sgn(I)B_i1∧ · · · ∧B_ij ∧A_ij+1∧ · · · ∧A_id

= sgn(I)

j

^

`=1

" _d X

k=1

βki_`Ak

#

∧

d

^

`=j+1

Ai_`

= sgn(I)

j

^

`=1

" j

X

r=1

βi_ri_`Ai_r

#

∧

d

^

`=j+1

Ai_`

= det (βi_ri_{`</sub>)<sup>j</sup><sub>r,`=1}

d

^

i=1

Ai

= det (β_{iri`</sub>)<sup>j</sup><sub>r,`=1}(−hV, E_d+1i)E₁∧ · · · ∧E_d. This shows that

d j

F_d−j^a (u,·) = Z

Ω

1{x∈ ·} X

|I|=j

det(β_I(x))H^d(dx), (3.7) whereβI(x) := (βij(x))_i,j∈Ifor a subsetI⊂ {1, . . . , d}.

Step 2. We prove that

X

|I|=j

det(βI(x)) =σj(D²u(x)) (3.8) forH^d-almost everyx∈ D²(u). Letx0∈ D²(u)be fixed so that the approximate tangent spaceTan^d(H^dxΓ(u),(x0,∇u(x0)))is ad-dimensional subspace ofR^d×R^dand

Tan := Tan^d(H^dxΓ(u),(x0,∇u(x0))) = lin{(Ai, Bi) :i= 1, . . . , d}.

Consider the linear mapϕ:R^d →R^ddefined byϕ:=π₂◦π⁻¹₁ , whereπ₁: Tan→R^d, (a, b)7→aandπ2 : Tan→R^d,(a, b)7→ b. Here we use thatπ1is surjective and hence also injective, sinceA:= (A1, . . . , Ad)is a basis ofR^d. The matrix ofϕwith respect to Ais given byM_ϕ^A = (βij)^d_i,j=1. Letλ1, . . . , λddenote the eigenvalues ofD²u(x0)with corresponding eigenvectorse1, . . . , ed, which form an orthonormal basisE := (e1, . . . , ed) ofR^d. We will prove that

Tan = lin{(ei, λ_ie_i) :i= 1, . . . , d}. (3.9) If this has been verified, then clearlyM_ϕ^E = diag(λ1, . . . , λd). But then (3.8) easily follows from the fact thatM_ϕ^E andM_ϕ^Ahave the same characteristic polynomial.

It remains to establish (3.9). It is sufficient to check that (e_i, λ_ie_i) ∈ Tan for i = 1, . . . , d. Let∇udenote a measurable choice of a subgradient field ofu(cf. [1, Theorem 8.1.3]), and letG(x) := (x,∇u(x)),x∈Ω. Then the inclusionG(Ω)⊂Γ(u)implies that Tan^d(H^dxG(Ω), G(x0))⊂Tan. (3.10) We will show that(e₁, λ₁e₁)(say) is contained in the set on the left-hand side of (3.10).

This will follow once we have established that Θ^∗d

H^dx(G(Ω)∩E(G(x0),(e1, λ1e1), )), G(x0)

>0 (3.11)

HESSIAN MEASURES OF CONVEX FUNCTIONS

for any∈(0,1); cf. [10, p. 252] for a definition of the setE(·,·,·)and [10, p. 181] for a definition of the upper densityΘ^∗d[·,·]. The proof of (3.11) requires some preparations.

Fix∈(0,1). Sincex0∈ D²(u), there is a functionω:R^d→R^dsuch that

∇u(x)− ∇u(x0)−D²u(x0)(x−x0) =kx−x0kω(x−x0), x∈Ω, and

x→xlim0

ω(x−x₀) = 0. Letδ()>0be chosen so thatx∈Ωand

kω(x−x0)k ≤ 8d

wheneverkx−x0k ≤δ(). Forρ∈(0, δ()]we define the pyramid (with apex removed) C_ρ,(x₀, e₁) :=

(

x0+t e1+

d

X

i=2

αiei

! :t∈

0, ρ

4(1 +λ)

,|αi| ≤ 1 1 +λ

2dfori= 2, . . . , d )

,

whereλ:= max{λ1, . . . , λd}. Letx∈Cρ,(x0, e1), i.e.

x=x0+t e1+

d

X

i=2

αiei

!

with t∈ 0, ρ(4(1 +λ))⁻¹ and|αi| ≤(1 +λ)⁻¹/(2d)fori= 2, . . . , d. Then

kx−x₀k=t (

1 +

d

X

i=2

α²_i )^1/2

≤t

1 + (d−1) ² 4d²

^1/2

≤2t≤ ρ 2(1 +λ), and consequently

kG(x)−G(x0)k= kx−x0k²+k∇u(x)− ∇u(x0)k²1/2

= kx−x0k²+kD²u(x0)(x−x0) +kx−x0kω(x−x0)k^21/2

≤ kx−x₀k(1 + (λ+ 1)²)^1/2

≤ kx−x0k2(1 +λ)≤ρ . Moreover, choosingr:=t⁻¹, we can estimate

kr(G(x)−G(x₀))−(e₁, λ₁e₁)k²

=k(r(x−x0)−e1, rD²u(x0)(x−x0) +rkx−x0kω(x−x0)−λ1e1)k²

=





d

X

i=2

αiei,

d

X

i=2

λiαiei+ (

1 +

d

X

i=2

α_i² )^1/2

ω(x−x0)





2

≤(d−1) ² 4d² +

d

X

i=2

|λi||αi|+√ 2

8d ²

+√

2 8d

²

≤ ² 4 +

d

X

i=2

2d+

4d ²

+ ² 32d² ≤

"

1 4+

3 4

²#

²≤².

ANDREA COLESANTI AND DANIEL HUG

Thus we have shown that, for >0andρ∈(0, δ()), Cρ,(x0, e1)

⊂ [

r>0

x∈Ω :G(x)∈B^2d(G(x₀), ρ),kr(G(x)−G(x₀))−(e₁, λ₁e₁)k ≤

=π1 G(Ω)∩B^2d(G(x0), ρ)∩E(G(x0),(e1, λ1e1), )

whereπ₁ : R^d×R^d →R^d is given byπ₁(x, y) := x. Letκ_d denote the volume of the Euclidean unit ball inR^d. Sinceπ₁is a contraction, we thus conclude that

Θ^∗d

H^dx(G(Ω)∩E(G(x0),(e1, λ1e1), )), G(x0)

≥lim sup

ρ↓0

H^d(Cρ,(x0, e1)) κ_dρ^d >0, which concludes the proof.

Hessian measures can be defined not just for convex functions, but also for semi-convex functions. Locally, a semi-convex function equals the sum of a convex function and a smooth function. A more detailed description and references are contained in [7]. In the following corollary, we describe the extension of Theorem 2 to this more general frame- work.

COROLLARY1. Theorem 2 remains true ifu: Ω→Ris semi-convex.

Proof. LetΩ⁰ be an open convex set whose closure is a compact subset ofΩ. Then there exists a constantC ≥0such thatv(x) := u(x) + (C/2)kxk²,x∈ Ω⁰, is convex.

Hence, overΩ⁰we have

Fd−j(u,·) =

j

X

i=0

j i

(−C)ⁱFd−j+i(v,·) ; (3.12) see Proposition 1.6 in [7]. Applying Theorem 2 tovin (3.12) and a well-known differenti- ation result for measures, we thus getH^d-almost everywhere inΩ⁰

d j

dF_d−j(u,·) dH^d =

j

X

i=0

d−j+i i

(−C)ⁱσ_j−i(D²v) =σj(D²u), which proves the assertion.

Theorem 2 will be used in the proof of our next result.

THEOREM3. LetΩbe an open convex subset ofR^d, and letu: Ω→Rbe a convex function. Assume that

F_d−1(u,·) H^d and ∆u∈L^p_loc(Ω) for somep≥2. Then, for everyk≤d−1such thatd−k≤p,

Fk(u,·) H^d and σ_d−k(D²u)∈L^q_loc(Ω), (3.13) for everyq≥1such that(d−k)q≤p.

Proof. Letφbe a standard mollifier inR^d, i.e.φ∈C^∞(R^d)has compact support in

HESSIAN MEASURES OF CONVEX FUNCTIONS

{x∈R^d:kxk ≤1}and

Z

R^d

φ(x)H^d(dx) = 1. For >0we define

φ(x) := 1 ^dφx

and

u(x) := (u∗φ)(x) = Z

R^d

u(y)φ(x−y)H^d(dy),

whereu(y) := 0fory /∈Ω. The functionuisC^∞and convex on compact subsets ofΩ, if >0is sufficiently small. Moreover, as→0⁺,uconverges uniformly touon compact subsets ofΩand, for everyk∈ {0,1, . . . , d},F_k(u,·)converges toF_k(u,·)in the sense of measures (in the vague topology); for a proof of the latter fact, see, for instance, [7, Theorem 1.1]. Letψbe a test function of classC^∞(Ω)with compact support inΩ. Using Theorem 2, we obtain

d lim

→0⁺

Z

ψ(x)F_d−1(u, dx) =d Z

ψ(x)F_d−1(u, dx) = Z

ψ(x)∆u(x)H^d(dx). On the other hand, by (1.2) and the divergence theorem,

d Z

ψ(x)F_d−1(u, dx) = Z

ψ(x)∆u(x)H^d(dx) = Z

u(x)∆ψ(x)H^d(dx) if >0is small enough. Hence, by the uniform convergence ofu, we get

d lim

→0⁺

Z

ψ(x)F_d−1(u, dx) = Z

u(x)∆ψ(x)H^d(dx). This shows that Z

ψ(x)∆u(x)H^d(dx) = Z

u(x)∆ψ(x)H^d(dx). (3.14) For an arbitrary pointy ∈Ω, letψ(x) :=φ(y−x). Hence, if >0is sufficiently small, thenψhas compact support inΩand (3.14) implies that

(∆u)(y) = (∆u)(y). (3.15)

Since by assumption∆u∈ L^p_loc(Ω),(∆u) →∆uinL^p_loc(Ω)(see [9, p. 123, Theorem 1]), and thus∆u→∆uinL^p_loc(Ω)as→0⁺.

Now we proceed to prove (3.13). Letk∈ {0, . . . , d−1}satisfyd−k≤p. Letη0⊂Ω be a Borel set withH^d(η0) = 0. Letη ⊂η0be a measurable subset with compact closure inΩ. SinceFd−1(u,·)is absolutely continuous, we obtain that∆u(x) = 0forH^dalmost everyx ∈ η. Moreover, since(∆u)^d−k is integrable, for any sufficiently small number δ >0, we can find an open setηδ ⊃ηwhose closure is a compact subset ofΩsuch that

Z

η_δ

(d⁻¹∆u(x))^d−kH^d(dx)≤δ . Subsequently, we will use Newton’s inequality in the form

d k

⁻¹

σ_d−k(D²u(x))≤(d⁻¹∆u(x))^d−k (3.16) for k ∈ {0, . . . , d−1}, which holds forH^d-almost everyx ∈ Ω. Applying the vague

ANDREA COLESANTI AND DANIEL HUG

convergence of the Hessian measures and (3.16), we get Fk(u, η)≤Fk(u, ηδ)≤lim inf

→0⁺ Fk(u, ηδ)

= lim inf

→0⁺

Z

η_δ

d k

−1

σ_d−k(D²u(x))H^d(dx)

≤lim inf

→0⁺

Z

η_δ

d⁻¹∆u(x)d−k

H^d(dx)

= Z

η_δ

d⁻¹∆u(x)^d−k

H^d(dx)≤δ .

We conclude that Fk(u, η) = 0, and hence by the monotone convergence thereom we obtainFk(u, η0) = 0.

The second part of the assertion (3.13) is now an immediate consequence of Theorem 2, the first part of the assertion (3.13) and (3.16).

As an easy consequence of Theorem 3 and its proof, we also have the following analogue of Satz 4.7 in [19].

COROLLARY2. LetΩbe an open convex subset ofR^d, and letu: Ω→Rbe a convex function. Then

Fd−1(u,·) H^d and ∆u≤Ca.e. inΩ, (3.17) for some constantC ≥ 0, if and only if there is a convex function v : Ω → R and a constantC⁰≥0such that

u(x) +v(x) =C⁰kxk², x∈Ω. (3.18) If one of these conditions is satisfied, thenFk(u,·) H^d with bounded density fork ∈ {0, . . . , d−2}.

Proof. First, assume that (3.17) is satisfied. LetΩ⁰be an open convex set with compact closure inΩ. The assumptions and equation (3.15) yield that∆u(x)≤Cforx∈Ω⁰and sufficiently small >0. Hence the Hessian matrix of the functionv:= (C/2)k · k²−u

onΩ⁰is positive semidefinite, and thereforevis convex onΩ⁰. Sinceu→uas→0⁺, the functionv:= (C/2)k · k²−uis also convex.

For the converse, we observe that by approximation with smooth functions, we get F_d−1(u+v,·) =F_d−1(u,·) +F_d−1(v,·)

for any convex functionsu, vonΩ. The assertion (3.17) now follows, since (3.18) implies thatF_d−1(u+v,·) = 2C⁰H^dand sinceF_d−1(v,·)≥0.

We can also derive Satz 4.7 in [19] by replacing the usual convolution employed above by the regularization of support functions described in Theorem 3.3.1 in [15]. A suitable modification of the proof of Corollary 2 then implies the following result which is equiva- lent to Weil’s theorem.

COROLLARY3. Letu:R^d→Rbe a support function. Then

F_d−1(u,·) H^d and ∆u(x)≤Ckxk⁻¹for a.e.xinR^d (3.19) and some constantC ≥0, if and only if there is a support functionv : R^d → Rand a

HESSIAN MEASURES OF CONVEX FUNCTIONS

constantC⁰≥0such that

u(x) +v(x) =C⁰kxk, x∈R^d.

Proof. First, let (3.19) be satisfied. For >0, we denote byuthe regularization ofu defined as in [15, Theorem 3.3.1]. Thenuis a support function of classC^∞onR^d\ {0}, u → uuniformly on compact subsets of R^d, and thus as in the proof of Theorem 3 we obtain that (∆u)(y) = (∆u)(y),y ∈ R^d. Hence, by the particular form of the regularization and by (3.19), we get, forx6= 0and≤1/2,

(∆u)(x) = Z

R^d

∆u(x+kxkz)ϕ(kzk)H^d(dz)

≤C Z

R^d

kx+kxkzk⁻¹ϕ(kzk)H^d(dz)

where ϕ denotes the mollifier defined in [15, Theorem 3.3.1] for the parameter . If ϕ(kzk)6= 0, thenkzk ≤≤1/2and hencekx+kxkzk ≥(1/2)kxk, i.e.

(∆u)(x)≤2Ckxk⁻¹, x∈R^d\ {0}. (3.20) From (3.20) we can conclude that the functionv:= 2Ck·k−uhas a positive semidefinite Hessian matrix onR^d\ {0}. Indeed, forx6= 0we have

D²v(x) = 2C 1 kxk

δ_ij−x_ix_j kxk²

d i,j=1

−D²u(x).

ThusD²v(x)has the eigenvalue0and furtherd−1nonnegative eigenvalues. This implies thatvis convex on convex subsets ofR^d\ {0}, which yields the convexity ofvonR^dby a continuity argument. Hencev:= 2Ck · k −uis a support function.

For the converse, we can proceed as in the proof of Corollary 2, since we also have F_d−1(u,{0}) = 0.

Similarly as Theorem 2, we can generalize Theorem 3 to semi-convex functions.

COROLLARY4. Theorem 3 remains true ifu: Ω→Ris semi-convex.

Proof. We use the same notation as in the proof of Corollary 1. By assumption, we haveFd−1(u,·) H^d and∆u ∈ L^p_loc(Ω). The following consideration can again be restricted to a relatively compact subsetΩ⁰ofΩ. A special case of (3.12) yields that

F_d−1(u,·) =F_d−1(v,·)−CF_d(v,·) =F_d−1(v,·)−CH^d;

henceF_d−1(v,·) H^d. Since∆v = ∆u+C ∈ L^p(Ω⁰), Theorem 3 implies that, for k≤dsuch thatd−k≤p,

Fk(v,·) H^d and σd−k(D²v)∈L^q(Ω⁰)

for everyq≥1such that(d−k)q≤p. For anyl≤dsuch thatd−l≤p, we thus get F_l(u,·) =

d−l

X

i=0

d−l i

(−C)ⁱF_l+i(v,·) H^d onΩ⁰. The required integrability follows from the relation

σ_d−l(D²u) =

d−l

X

i=0

l+i i

(−C)ⁱσ_d−l−i(D²v),

ANDREA COLESANTI AND DANIEL HUG

which holdsH^d-almost everywhere inΩ⁰, sinceσ_d−l−i(D²v)∈L^q(Ω⁰)fori≤d−land (d−l)q≤p.

Theorem 3 can be used to give a simplified proof (on the basis of the results for Hessian measures) of a theorem due to Weil [19], which concerns surface area measures of convex bodies. A corresponding result for curvature measures has been deduced recently in [12]

from the one for surface area measures. However, it does not seem to be possible to derive the result for curvature measures directly from Theorem 3. In the following, we write R_j(K,·)for the Radon-Nikodym derivative of thejth surface area measureS_j(K,·)of a convex bodyK ∈ K^d_o with respect to the(d−1)-dimensional Hausdorff measureH^d−1 onS^d−1. We state the following result in a global form, but it is clear from the arguments presented that all statements can be localized as in [19].

THEOREM(Weil). LetK∈ K^d_osatisfy

S1(K,·) H^d−1 and R1(K,·)∈L^p(S^d−1) for somep≥2. Then, for everyk∈ {2, . . . , d−1}such thatk≤p,

S_k(K,·) H^d−1 and R_k(K;·)∈L^q(S^d−1) for everyq≥1such thatqk≤p.

Proof. The assertion of the theorem follows from Theorem 3, once we have shown that the conditions

(a) Fj(hk,·) H^dandσd−j(D²hK)∈L^p_loc(R^d) and

(b) S_d−j(K,·) H^d−1andR_d−j(K,·)∈L^p(S^d−1) are equivalent forj∈ {1, . . . , d−1}.

First, assume that (b) holds. Letη ⊂ R^d be a Borel set. Then, using Lemma 1 and introducing polar coordinates, we get

dF_j(h_K, η) =j Z∞

0

Z

S^d−1

1{rν∈η}r^j−1R_d−j(K, ν)H^d−1(dν)dr

=j Z

R^d

1{x∈η}kxk^j−dRd−j(K, x/kxk)H^d(dx), which shows thatFj(hK,·) H^dand

σ_d−j(D²hK(x)) =kxk^j−d d−1

j

R_d−j(K, x/kxk)

forH^d-almost everyx∈R^d. Now the required integrability property follows easily.

Conversely, assume that (a) is satisfied. Letω⊂S^d−1be a Borel set and0< a < b <

∞. Then, applying Lemma 1, introducing polar coordinates and using the fact thatD²h_K is positively homogeneous of degree−1, we get that

S_d−j(K, ω) = d b^j−a^j

Z

R^d

1{kxk ∈[a, b], x/kxk ∈ω}

d j

−1

σ_d−j(D²hK(x))H^d(dx)

= d−1

j −1Z

ω

σ_d−j(D²h_K(ν))H^d−1(dν), from which (b) follows as above.

HESSIAN MEASURES OF CONVEX FUNCTIONS

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Andrea Colesanti

Dipartimento di Matematica Universit`a degli Studi di Firenze Viale Morgagni 67/A

50134 Firenze Italy

andrea.colesanti@bb.math.unifi.it

Daniel Hug

Mathematisches Institut Albert-Ludwigs-Universit¨at Eckerstr. 1

D-79104 Freiburg Germany

daniel.hug@math.uni-freiburg.de