# Extremal rank-one convex integrands and a conjecture of Šverák

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## Abstract

We show that, in order to decide whether a given probability measure is laminate, it is enough to verify Jensen’s inequality in the class of extremal non-negative rank-one convex integrands. We also identify a subclass of these extremal integrands, consisting of truncated minors, thus proving a conjecture made by Šverák (Arch Ration Mech Anal 119(4):293–300, 1992).

## Mathematics Subject Classification

Primary 49J45 Secondary 46N10## 1 Introduction

Since its introduction in the seminal work of Morrey [32], quasiconvexity has played an important role not just in the Calculus of Variations [11, 13, 21, 40] but also in problems from other areas of mathematical analysis, for instance in the theory of compensated compactness [35, 46]. Nonetheless, this concept is still poorly understood and has been mostly studied in relation with polyconvexity and rank-one convexity, which are respectively stronger and weaker notions that are easier to deal with (we refer the reader to Sect. 2 for terminology and notation). An outstanding open problem in the area is Morrey’s problem, which is the problem of deciding whether rank-one convexity implies quasiconvexity, so that the two notions coincide. A fundamental example [44] of Šverák shows that this implication does not hold in dimensions \(3\times 2\) or higher and, more recently, Grabovsky [16] has found a different example in dimensions \(8\times 2\) which moreover is 2-homogeneous. The problem in dimensions \(2\times 2\), in particular, remains completely open, but in the last two decades evidence towards a positive answer in this case has been piling up [2, 14, 17, 24, 25, 33, 37, 38, 41].

^{1}in the space of \(2\times 2\) symmetric matrices are quasiconvex. In this direction, Šverák introduced in [43] new quasiconvex integrands, which were later generalized in [15]. For any \(n\times n\) symmetric matrix

*A*, these integrands are defined by

*index*of a matrix is the number of its negative eigenvalues. We also note that the integrand \(F_0\) is sometimes called \(\det ^{++}\) in the literature, since its support is the set of positive definite matrices. These integrands have played an important role in studying other problems related to the Calculus of Variations, for instance in building counterexamples to the regularity of elliptic systems [34] or in the computation of rank-one convex hulls of compact sets [45].

In order to understand Šverák’s motivation for considering these integrands it is worth making a small excursion into classical convex analysis. Given a real vector space \({\mathbb {V}}\) and a convex set \(K\subset {\mathbb {V}}\), one can define the set of *extreme points* of *K* as the set of points which are not contained in any open line segment contained in *K*. In general the set of extreme points might be very small: this is what happens, for instance, when the set is a convex cone \(C\subset {\mathbb {V}}\), since in this situation all non-zero vectors are contained in a ray through zero. However, if we can find a convex base *B* for *C*, then we note that such a ray corresponds to a unique point in *B*. If this is an extreme point of *B* then we say that we have an *extremal ray*.

We are interested in the extremal rays of the cone *C* of rank-one convex integrands. This cone has the inconvenient feature that it is not *line-free*: there is a set of elements \(v\in {\mathbb {V}}\) such that, for any \(c\in C\) and any \(t\in {\mathbb {R}}\), the point \(c+tv\) is in *C*; this set is precisely \(C\cap (-C)\). In turn, it is quite clear that this is the set of rank-one affine integrands. A reasonable way of disposing of rank-one affine integrands is by demanding non-negativity from all integrands from *C*. This leads us to the definition of extremality considered by Šverák: we say that a non-negative rank-one convex integrand *F* is *extremal* if, whenever we have \(F=E_1+E_2\) for \(E_1, E_2\) non-negative rank-one convex integrands, then each \(E_i\) is a non-negative multiple of *F*. A weaker notion of extremality was introduced by Milton in [31] for the case of quadratic forms (see also [18]) but we shall not discuss it further here.

^{2}:

### Theorem 1.1

Given a minor \(M:{\mathbb {R}}^{n\times n}\rightarrow {\mathbb {R}}\), let \(M^\pm \) be its positive and negative parts. Then \(M^\pm \) are extremal non-negative rank-one convex integrands in \({\mathbb {R}}^{n\times n}\).

For \(k=0,\dots ,n\), the integrands \(F_k:{\mathbb {R}}^{n\times n}_{\text {sym}}\rightarrow {\mathbb {R}}\), are extremal non-negative rank-one convex integrands in \({\mathbb {R}}^{n\times n}_{\text {sym}}\).

As a main tool we use the fact that, on a connected open set, a rank-one affine integrand is an affine combination of minors; this is proved by localizing the arguments from Ball–Currie–Olver [5] concerning the classification of Null Lagrangians.

The importance of extreme points in convex analysis has to do with the Krein–Milman theorem, which states that the closed convex hull of the set of extreme points of a compact, convex subset *K* of a locally convex vector space is the whole set *K*—informally, this means that the set of extreme points is a set of “minimal information” needed to recover *K*. However, Klee [26] showed that Krein–Milman theorem is generically true for trivial reasons: if we fix an infinite-dimensional Banach space and we consider the space of its compact, convex subsets (which we can equip with the Hausdorff distance so that it becomes a complete metric space), then for almost every compact convex set *K* its extreme points are dense in *K*. Here we mean “almost every” in the sense that the previous statement is false only in a meagre set. Despite this disconcerting result, the situation can be somewhat remedied with the help of Choquet theory, which roughly states that, under reasonable assumptions, an arbitrary point in *K* can be represented by a measure carried in the set of its extreme points. For precise statements and much more information concerning Choquet theory we refer the reader to the lecture notes [39] or to the monograph [28] and for Krein–Milman-type theorems for semi-convexity notions see [23, 27, 29].

### Theorem 1.2

*Q*. A Radon probability measure \(\nu \) supported on the interior of

*Q*is a laminate if and only if

*g*which are extreme points of the convex set

We note that the summation condition is simply a normalization which corresponds to fixing a base of the cone of non-negative rank-one convex integrands on *Q*.

Our interest in extremal integrands was ignited by the work [3] of Astala–Iwaniec–Prause–Saksman, where it was shown that an integrand known as Burkholder’s function is extreme in the class of homogeneous, isotropic, rank-one convex integrands; in fact, this integrand is also the least integrand in this class, in the sense that no other element of the class is below it at all points. The relevance of this fact is readily seen: from standard results about quasiconvex envelopes it follows immediately that the Burkholder function is either quasiconvex everywhere or quasiconvex nowhere. Burkholder’s function was found in the context of martingale theory by Burkholder [9, 10] and was later generalized to higher dimensions by Iwaniec [19]. This remarkable function is a bridge between Morrey’s problem and important problems in Geometric Function Theory [1, 20] and we refer the reader to the very interesting papers [2, 3, 19] and the references therein for more details in this direction.

Finally, we give a brief outline of the paper. Section 2 contains standard definitions, notation and briefly recalls some useful facts for the reader’s convenience. Section 3 comprises results concerning improved homogeneity properties of rank-one convex integrands which vanish on isotropic cones. Section 4 is devoted to the proof of Theorem 1.1. Finally, Sect. 5 elaborates on the relation between Choquet theory and Morrey’s problem and we prove Theorem 1.2.

## 2 Preliminaries

In this section we will gather a few definitions and notation for the reader’s convenience. The material is standard and can be found for instance in the excellent references [12] and [33].

*E*is

*polyconvex*if

*E*(

*A*) is a convex function of the minors of

*A*, see [6]. If

*E*is locally integrable, we say that

*E*is

*quasiconvex*if there is some bounded open set \(\Omega \subset {\mathbb {R}}^n\) such that, for all \(A\in {\mathbb {R}}^{m\times n}\) and all \(\varphi \in C^{\infty }_0(\Omega ,{\mathbb {R}}^n)\),This notion was introduced in [32] and generalized to higher-order derivatives by Meyers in [30]. The case of higher derivatives was also addressed in [5], where it was shown that quasiconvexity is not implied by the corresponding notion of rank-one convexity if \(m>2\) and \(n\ge 2\). The integrand

*E*is

*rank-one convex*if, for all matrices \(A, X\in {\mathbb {R}}^{m\times n}\) such that

*X*has rank one, the function

*E*is rank-one convex if, whenever \((A_i, \lambda _i)_{i=1}^N\) satisfy the \((H_N)\) conditions (c.f. [12, Def. 5.14]), we have

*prelaminate*. It is also clear how to adapt the definition of rank-one convexity to the more general situation where

*E*is defined on an open set \(\mathcal {O}\subset {\mathbb {R}}^{m\times n}\). Finally, \(E:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) is

*separately convex*if, for all \(x\in {\mathbb {R}}^d\) and all \(i=1,\dots , d\), the function \(t\mapsto E(x+t e_i)\) is convex; we denote by \(e_1,\dots , e_d\) the standard basis of \({\mathbb {R}}^d\).

*E*quasiconvex \(\Leftarrow E\) rank-one convex fails if \(n\ge 2, m\ge 3\). The case \(n\ge m=2\) is the content of Morrey’s problem.

*n*-homogeneous (in fact, they are

*n*-homogeneous if

*n*is even) and isotropic. A generic integrand \(E:{\mathbb {R}}^{n\times n}\rightarrow {\mathbb {R}}\) is said to be

*p*-

*homogeneous*for a number \(p\ge 1\) if

*positively p-homogeneous*if the same holds only for \(t>0\). The integrand

*E*is

*isotropic*if it is invariant under the left– and right–\(\text {SO}(n)\) actions, that is,

*singular values*\({{\widetilde{\sigma }}}_1(A)\ge \dots \ge {{\widetilde{\sigma }}}_n(A)\ge 0\) of a matrix

*A*are the eigenvalues of the matrix \(\sqrt{AA^T}\). We shall consider the

*signed*singular values \(\sigma _j(A)\), which are defined by

### Theorem 2.1

*E*is isotropic,

*conformal matrices*while \({\mathbb {R}}^{2\times 2}_{\text {aconf}}\) corresponds to the

*anticonformal matrices*; these are the matrices that are scalar multiples of orthogonal matrices and have respectively positive and negative determinant. This decomposition is particularly useful for us because the singular values of

*A*satisfy the identities

*E*is isotropic, \(E(A)=E(|A^+|,|A^-|)\). In particular, the above formulae yield

*Burkholder’s function*. This function can be defined in any real or complex Hilbert space with the norm \(\Vert \cdot \Vert \) by

*p*-homogeneous. If the Hilbert space where \(B_p\) is defined is \({\mathbb {C}}\), the zig-zag convexity of \(B_p\) implies that the Burkholder function \(B_p:{\mathbb {C}}\times {\mathbb {C}}\rightarrow {\mathbb {R}}\) is rank-one convex. Since we are interested in non-negative integrands, we will also deal with the integrand \(B_p^+\equiv \max (B_p,0)\), which is also rank-one convex. Moreover, \(B_2^+=\det ^+\), so \(B_p^+\) can be seen as a “\(\det ^+\)-type integrand”, in the sense that it is rank-one convex, isotropic and vanishes on some cone

## 3 Homogeneity properties of a class of rank-one convex integrands

### Lemma 3.1

*E*is non-positive on \(\mathcal {C}_a\). Define

### Proof

We now specialise the lemma to two important situations, in which one can say more. Let us first assume that \(k=0\).

### Proposition 3.2

Let \(E:{\mathbb {C}}\times {\mathbb {C}}\rightarrow {\mathbb {R}}\) be rank-one convex, positively *p*-homogeneous for some \(p\ge 1\) and not identically zero. If there is some \(a\ge 1\) such that \(E=0\) on \(\mathcal {C}_a\) then \(p\ge \frac{1}{a} + 1\).

### Proof

To finish the proof it suffices to prove the claim. Take an arbitrary \(z\in S^1\) and take any rank-one line segment starting at (*z*, 0) and having the other end-point in \(\mathcal {C}_a\); such a line must intersect \(\mathcal {C}_1\), since \(a>1\), say at \(P_z\). Note that \(E(P_z)\ge 0\), since the function \(t\mapsto E(t P_z)= t^p E(P_z)\) is convex. We conclude that \(E(z,0)\ge 0\) with equality if and only if \(E(P_z)=0\), in which case *E* is identically zero along the entire rank-one line segment.

*E*is identically zero, so let us make this assumption. Then, from the previous discussion, we see that

*E*is identically zero in the “outside” of \(\mathcal {C}_a\), i.e. in

*P*in the interior of \(\mathcal {C}_a\), there is a rank-one line segment through

*P*with both endpoints, say \(P_1, P_2\), in \(\mathcal {C}_a^+\); this is the case because we assume \(a>1\). But

*E*is zero in a neighbourhood of \(P_i\) and since it is convex along the rank-one line segment \([P_1,P_2]\) we conclude that it is also zero at

*P*. \(\square \)

We remark that, in one dimension, the only homogeneous extreme convex integrands are linear (c.f. Proposition 5.3) while, from the results of Sect. 4, for \(n>1\) there are extremal rank-one convex integrands in \({\mathbb {R}}^{n\times n}\) which are positively *k*-homogeneous for any \(k\in \{1,\dots , n\}\). It would be interesting to know whether there are extremal homogeneous integrands with other degrees of homogeneity, or whether there is an upper bound for the order of homogeneity of such integrands.

If we set \(a=1\) in Lemma 3.1, so \(\mathcal {C}_1=\{\det =0\}\), we see that \(h_1(t,k)=t^{-2}\) and we find the estimate \(t^2 E(A)\le E(tA)\) for \(t\ge 1\). In fact, this holds in any dimension, and the proof is a simple variant of the proof of Lemma 3.1.

### Lemma 3.3

*E*satisfies

### Proof

*B*to get

*A*is diagonal, so there are real numbers \(\sigma _j\) such that \(A=\text {diag}(\sigma _1, \dots , \sigma _n)\). Let \(A_0\equiv A\) and define, for \(1\le j \le n\),

*E*yields

*A*is not diagonal, we consider the singular value decomposition of Theorem 2.1, i.e. \(A=Q\Sigma R\) where \(Q,R\in \text {SO}(n)\) and \(\Sigma =\text {diag}(\sigma _1, \dots , \sigma _n)\). We see that (3.2) can be rewritten as

*Q*and

*R*, we get

*A*is still a prelaminate. For this, we use the following elementary fact:

As a simple consequence of the lemma, we find a rigidity result for decompositions of positively *n*-homogeneous integrands.

### Proposition 3.4

Let \(E_1, E_2:{\mathbb {R}}^{n\times n}\rightarrow {\mathbb {R}}\) be rank-one convex integrands which are non-positive on \(\{\det =0\}\) and assume there is some positively *n*-homogeneous integrand *F* such that \(F=E_1+E_2\). Then each \(E_i\) is positively *n*-homogeneous.

### Proof

*n*-homogeneous they must be equal in the whole set

*U*and so \(E_i=E_i^h\) in

*U*. An identical argument establishes equality in the complement of

*U*. \(\square \)

### Remark 3.5

The proofs of Lemma 3.3 and Proposition 3.4 are fairly robust. In particular, a similar statement holds if the integrands \(E_i\) are defined in \({\mathbb {R}}^{n\times n}_{\text {sym}}\) instead of \({\mathbb {R}}^{n\times n}\). Indeed, \({\mathbb {R}}^{n\times n}_{\text {sym}}\) is the set of (real) matrices that can be diagonalized by rotations. Thus, the prelaminate built in the proof of Lemma 3.3 has support in \({\mathbb {R}}^{n\times n}_{\text {sym}}\) if *A* is symmetric: for the nondiagonal case, one can take \(R=Q^{-1}\).

## 4 Proof of extremality for truncated minors

This section is dedicated to the proof of Theorem 1.1. Although truncated minors are not linear along rank-one lines, they are piecewise linear along such lines. For this reason, it will be useful to have at our disposal the classification of rank-one affine integrands, which is due to Ball [6] in dimensions three or lower, Dacorogna [12] in higher dimensions and also Ball–Currie–Olver [5] in the case of higher order quasiconvexity. Given an open set \(\mathcal {O}\subset {\mathbb {R}}^{n\times n}\) and an integrand \(E:\mathcal {O}\rightarrow {\mathbb {R}}\) we say that *E* is *rank-one affine* if both *E* and \(-E\) are rank-one convex; such integrands are also often called *Null Lagrangians* or *quasiaffine*.

### Theorem 4.1

Let \(\mathcal {O}\subset {\mathbb {R}}^{n\times n}\) be a connected open set and consider a rank-one affine integrand \(E:\mathcal {O}\rightarrow {\mathbb {R}}\). Then *E*(*A*) is an affine combination of the minors of *A*.

*A*and let \(\tau (n)\equiv (2n)!/(n!)^2\) be its length. There is a constant \(c\in {\mathbb {R}}\) and a vector \(v\in {\mathbb {R}}^{\tau (n)}\) such that

This theorem is essentially a particular case of [5, Theorem 4.1], the only difference being that in this paper the authors deal only with integrands defined on the whole space. We briefly sketch how to adapt their proof to our case. The first result needed is the following:

### Lemma 4.2

In particular, when \(\mathcal {O}\) is connected, any continuous rank-one affine integrand *E* is a polynomial of degree at most *n*.

We remark that our proof is very similar to the one in [6, Theorem 4.1].

### Proof

*E*is rank-one affine if and only if

*E*is rank-one affine and fix some point \(A\in \mathcal {O}\). Define the 2

*k*-tensor \(T:({\mathbb {R}}^n)^{2k}\rightarrow {\mathbb {R}}\) by

*E*is rank-one affine,

*T*is alternating. This follows from the following claim: if \(w_j=w_l\) for some \(j\ne l\), then

*E*is smooth, let us take \(w_1, \dots , w_k\) linearly dependent, so we can suppose for simplicity that \(w_k=w_1+\dots + w_{k-1}\). Then

*T*is linear and alternating. The last statement of the lemma follows by observing that the first part implies that \(D^{n+1}E(A)=0\) for all \(A\in \mathcal {O}\).

*E*is merely continuous, let \(\rho \) be the standard mollifier and let \(\rho _\varepsilon (A)=\varepsilon ^{-n^2}\rho (A/\varepsilon )\) for \(\varepsilon >0\). Fix \(A\in \mathcal {O}\) and find an \(\varepsilon >0\) such that \(\text {dist}(A,\partial \mathcal {O})>\varepsilon \). Then \(E_\varepsilon \equiv \rho _\varepsilon *E\) is smooth and rank-one affine and hence

Using the lemma, we see that in order to prove the theorem it suffices to consider rank-one affine integrands which are homogeneous polynomials, so let us take such an integrand *E* which is a homogeneous polynomial of some degree *k*. Given any \(A\in \mathcal {O}\), the total derivative \(D^k E(A)\) is a symmetric *k*-linear function \(D^k E(A):({\mathbb {R}}^{n\times n})^k\rightarrow {\mathbb {R}}\); we remark that this operator has as domain the whole matrix space and not just a subset of it. There is an isomorphism between the space of *k*-homogeneous rank-one affine integrands and the space of symmetric *k*-linear functions \(({\mathbb {R}}^{n\times n})^k\rightarrow {\mathbb {R}}\) and the proof in [5] is unchanged in our case.

We recall that a generic symmetric rank-one matrix is of the form \(c v\otimes v\) for some \(v\in {\mathbb {R}}^n\) with \(|v|=1\) and some \(c \in {\mathbb {R}}\). Hence, we have the following analogue of Lemma 4.2:

### Lemma 4.3

From this we deduce, by the same arguments as in the situation above, the following result:

### Theorem 4.4

*M*, let \(\mathcal {O}_M\equiv \{M>0\}\). Moreover, each \(F_k\) has support in the set

### Lemma 4.5

For any minor *M* the set \(\mathcal {O}_M\) is connected. Moreover, for \(k=0,\dots , n\), the sets \(\mathcal {O}_k\) are connected.

### Proof

*M*be an \(s\times s\) minor. Let us make the identification

For the second part, note that the set \(\mathcal {O}_k\) is the set of matrices *A* for which there is some \(Q\in \text {SO}(n)\) and some diagonal matrix \(\Lambda =\text {diag}(a_1, \dots , a_k, b_1, \dots , b_{n-k})\), where \(a_i<0\) and \(b_j>0\), such that \(QAQ^T=\Lambda \). Clearly the set of \(\Lambda \)’s with this form can be connected to \(\text {diag}(-I_k, I_{n-k})\) by a path in \(\mathcal {O}_k\); here \(I_l\) is an \(l\times l\) identity matrix. Hence it suffices to prove that there is a continuous path in \(\mathcal {O}_k\) connecting \(A=Q\Lambda Q^T\) to \(\Lambda \). Such a path is given by \(A(t)=Q(t)A Q(t)^T\), where \(Q:[0,1]\rightarrow \text {SO}(n)\) is a continuous path with \(Q(0)=I, Q(1)=Q\). \(\square \)

We are finally ready to prove the extremality of truncated minors and of Šverák’s integrands.

### Proof of Theorem 1.1

*M*be a minor and let \(E_1, E_2:{\mathbb {R}}^{n\times n}\rightarrow [0,\infty )\) be rank-one convex integrands such that \(M^+=E_1+E_2\). For concreteness, let us say

*j*-th order minors of

*A*(this is denoted by \(\text{ adj }_j(A)\) in [12]).

*s*and some minor \(M'\) of order

*s*, there is a matrix

*A*such that \(M'\) is the only minor of order

*s*that does not vanish at

*A*. Indeed, if

*A*whose only non-zero entries are the entries \(a_{i'_\alpha j'_\alpha }\) for \(\alpha \in \{1,\dots , s\}\) and set these entries to one, so \(M'(A)=1\). Since all other entries of

*A*are zero we see that all other minors of order

*s*vanish at

*A*. Note, moreover, that

*A*has rank

*s*.

The previous observation, applied with \(s=k\) and \(M=M'\), shows that for \(A\in \overline{\mathcal {O}_M}\) we have \(v_i^k\cdot {\mathbf {M}}_k(A)= \lambda _i M(A)\), where \(\lambda _i \in {\mathbb {R}}\) is the entry of \(v_i^k\) corresponding to *M*. We now prove that all the vectors \(v_i^j, j\ne k\), are zero.

*j*, say \(M'=e_\alpha \cdot {\mathbf {M}}_j\) for some \(\alpha \), there is an

*A*with \(\text{ rank }(A)=j\) so that \(M'\) is the only minor of order

*j*that does not vanish at

*A*. Since

*A*has rank

*j*all of its \((j+1)\times (j+1)\) minors vanish and, in particular, \(M(A)=0\) and hence \(A\in \overline{\mathcal {O}_M}\). Since

*j*is the lowest integer for which \(v_i^j\ne 0\) we have

*j*, say \(M'=e_\alpha \cdot {\mathbf {M}}_j\) for some \(\alpha \), there is an

*A*such that \(M'\) is the only minor of order

*j*that does not vanish at

*A*; moreover, by flipping the sign of the \(i_1\)-th row of

*A*, if need be, we can assume that \(A \in \overline{\mathcal {O}_M}\). Since \(j>k\) is the highest integer for which \(v_i^j\ne 0\), by computing

*t*, the sign of \((v_i^j)_\alpha M'(A)\); hence \((v_i^j)_\alpha =0\) for \(i=1,2\). Moreover, since \(\alpha \) was chosen arbitrarily we have \(v_i^j=0\) and we find a contradiction; thus \(j=k\).

*J*changes the sign of the \(i_1\)-th row.

*n*-homogeneous and therefore \(E_i= \alpha _i \det \) in \(\mathcal {O}_k\), where \(\alpha _i\) is the last entry of \(v_i\). Since \(E_i\ge 0\) we must have, by possibly changing the sign of \(\alpha _i\), \(E_i=\alpha _i |\det |\) in \(\mathcal {O}_k\). Moreover, \(E_i=0\) outside \(\mathcal {O}_k\), and so indeed \(E_i=\alpha _i F_k\) as wished. \(\square \)

## 5 Choquet theory and Morrey’s problem

The main tool of this section is the following powerful result:

### Theorem 5.1

*K*be a metrizable, compact, convex subset of a locally convex vector space

*X*. For each \(f\in K\) there is a regular probability measure \(\mu \) on

*K*which is supported on the set \(\text{ Ext }(K)\) of extreme points of

*K*and which

*represents*the point

*f*: for all \(\varphi \in X^*\),

For a proof see, for instance, [39, §3]. We note that in general—and this is also the case in our situation—the representing measure is not unique. In order to apply this theorem to \(\mathcal {B}_d^\Box \), we need to show that this set is metrizable and this can be done by using a simple result from point-set topology; for a proof see, for instance, [28, Lemma 10.45].

### Lemma 5.2

Let *K* be a compact Hausdorff space. Then *K* is metrizable if and only if there is a countable family of continuous functions on *K* which separates points.

In our situation, it is easy to see that such a family exists: indeed, let \(x_n\in Q_d\) be a countable set of points which is dense in \(Q_d\) and consider the evaluation functionals \(\varepsilon _{x_n}:f\mapsto f(x_n)\) on \(\mathcal {B}_d^\Box \). These functionals are continuous on \(\mathcal {B}_d^\Box \) and separate points, since all elements of \(\mathcal {B}_d^\Box \) are continuous real-valued functions on \(Q_d\). Therefore the lemma implies that \(\mathcal {B}_d^\Box \) is metrizable, and hence Choquet’s theorem yields:

### Proposition 5.3

Fix \(\Box \in \{\text{ c },\text{ qc },\text{ rc },\text{ sc }\}\) and let \(\nu \) be a regular probability measure in *Q*. The measure \(\nu \) satisfies \(f({\overline{\nu }} )\le \langle \nu ,f \rangle \) for all \(f\in \mathcal {C}_d^\Box \) if and only if \(g({\overline{\nu }} )\le \langle \nu ,g \rangle \) for all \(g\in \text{ Ext }(\mathcal {B}_d^\Box )\).

### Proof

Theorem 1.2 follows easily from Proposition 5.3. For the reader’s convenience, we restate the theorem here:

### Theorem 5.4

### Proof

*non-negative*rank-one convex integrands then it holds for any rank-one convex integrand: given any such

*f*, one can consider the new integrand

*f*, being continuous, is bounded by below on \(Q_d\).

Therefore, from Proposition 5.3, the theorem follows once we show that any rank-one convex integrand \(g:Q_d\rightarrow [0,\infty )\) can be extended to a rank-one convex integrand \(f:{\mathbb {R}}^{n\times n}\rightarrow {\mathbb {R}}\) with \(g=f\) in the support of \(\nu \). This is a standard result, see [42]. \(\square \)

We end this section with some cautionary comments concerning the previous results. In the one dimensional case, where all the above cones coincide, the extreme points are quite easy to identify; the oldest reference we found where this problem is discussed is [7], but see also [28, §14.1].

### Proposition 5.5

In higher dimensions the various cones are different. In the case of convex integrands, the set of extreme points of \(\mathcal {C}_d^\text{ c }\) for \(d>1\) is very different from the one-dimensional case, since it is dense in this cone.

### Theorem 5.6

Any finite continuous convex function on a convex domain \(U\subset {\mathbb {R}}^d\) can be approximated uniformly on convex compact subsets of *U* by extremal convex functions.

This result was proved by Johansen in [22] for \(d=2\) and then generalized to any \(d>1\) in [8]. In these papers the set of extremal convex functions is not fully identified, but it is shown that there is a sufficiently large class of extremal convex functions which approximate any given convex function well: these are certain *polyhedral functions*, i.e. functions of the form \(f=\max _{1\le i \le k} a_i\) for some affine functions \(a_1, \dots , a_k\). This disturbing situation, however, is not too unexpected given the result of Klee [26] already mentioned in the introduction. I do not know whether a similar statement holds for the cones \(\mathcal {C}_{n\times m}^\text{ qc }\) and \(\mathcal {C}_{n\times m}^\text{ rc }\).

## Footnotes

- 1.
We will refer to real-valued functions defined on a matrix space as

*integrands*; this terminology is standard in the Calculus of Variations literature. - 2.
In fact, Šverák only conjectures extremality in the cone of quasiconvex integrands, so our results are in this sense slightly stronger than his conjecture.

## Notes

### Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. I thank my supervisor Jan Kristensen for guidance throughout the process of obtaining these results as well as for carefully reading and making corrections to the drafts. I would also like to thank István Prause, Rita Teixeira da Costa and Lukas Koch for helpful discussions and comments.

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