n-polyconvex hulls and the envelope of the distance function

.

(ii) Let P_n⁰f = f and for k ∈ N

P_n^k+1f (F ) = inf







τ (n)+1

X

i=1

λ_iP_n^kf (F_i) : λ ∈ Λ_{τ (n)+1},

V simple rank-n, Fi ∈ F + V, T (F ) =

τ (n)+1

X

i=1

λiT (Fi)







. (4.54)

Then Pnf = lim_k→∞P_n^kf = inf_k∈NP_n^kf .

Similar to Theorem 4.13 the proof is absolutely analogous and will be omitted.

4.4.2. n-polyconvex hulls and the envelope of the distance function

The first definitions and results in this section will consider hulls from an intersectional approach, i.e. by considering points from a given set and adding all those points that can be obtained by relevant convex combinations. In the latter part of this section we will investigate different kind of hulls arising from a separational approach, i.e. all points that cannot be separated from the set by finite n-polyconvex functions. Following [40]

for D-convexity we will call these hulls functional n-polyconvex hulls.

Let us begin with the hulls from an intersectional point of view.

Definition 4.50. Let K ⊆ R^d×D. Then we denote by P_n^∩K the smallest n-polyconvex set that contains K and refer it as the intersectional n-polyconvex hull of K.

Then the following theorem corresponds to its counterpart for envelopes, Theorem 4.49.

Theorem 4.51. Let K ⊆ R^d×D. Let P_n⁰K := K and define inductively P_n^k+1K = {F ∈ R^d×D : ∃V ⊆ R^d×D simple rank-n, F_i ∈ P_n^kK ∩ (F + V ),

i = 1, . . . , τ (n) + 1, λ ∈ Λ_{τ (n)+1}, T (F ) =

τ (n)+1

X

i=1

λ_iT (F_i)}.

Then P_n^∩K =S

k∈NP_n^kK. Moreover, if K is open then P_n^∩K is open.

Remark 4.52. (i) For the intersectional rank-one convex hull (which corresponds to P₁^∩K) this kind of hull is also often referred to as the lamination convex hull.

(ii) For n = d ∧ D, i.e the polyconvex case, we find that P_d∧D¹ K = P_d∧D^∩ K and so no lamination is necessary.

The proof of the above theorem is analogous to the envelope version and again, we omit most of it here. However, since the proof of the openness of the rank-one convex hull of an open set is only hinted at, we will discuss this part in more detail.

Proof of openness of P_n^∩K for open K in Theorem 4.51. Let K be open. We simply prove that then P_n¹K must be open and the whole statement follows by induction.

Let F ∈ P_n¹K. Then there exist a simple rank-n subspace V and matrices Fi, i = 1, . . . , k, k ≤ τ (n) + 1, belonging to K ∩ (F + V ), and λ ∈ Λ_k such that T (F ) =Pk

i=1λ_iT (F_i).

Since K is open and k finite there exists ε > 0 such that B_ε(F_i) ⊆ K for all i = 1, . . . , k.

We claim that then Bε(F ) ⊆ P_n¹K, proving that P_n¹K is indeed open. Let eF ∈ Bε(F ).

Then define eF_i = F_i+ eF − F ∈ K. From Lemma A.8 of Appendix A.4 we have that also T ( eF ) =Pk

i=1λiT ( eFi) and so eF ∈ P_n¹K.

The proof for the rank-one convex case can be found after Theorem 7.17 in [20]. The same applies to the following, cf. [20, Prop. 7.22].

Proposition 4.53. Let K ⊆ R^d×D and χ_K its characteristic function. Then P_nχ_K = χ_P∩

nK

where P_nχ_K is the n-polyconvex envelope of the characteristic function χ_K as defined in Definition 4.48.

The proof of this proposition is simple and in fact it holds that P_n^kχ_K= χ_Pk

nK for each k ∈ N, which can be proved by induction. Next we observe that the preservation of the convexity properties of a set is maintained for the interior of the set also in the case of n-polyconvexity. However, as to be expected, this is not true for the closure.

Proposition 4.54. (i) Let K ⊆ R^d×D be n-polyconvex. Then int K is n-polyconvex.

(ii) There exists K ⊆ R^2×2 such that K is polyconvex (and hence n-polyconvex for any n), but K is not separately convex (and hence not n-polyconvex for any n).

The proof of ‘(i)’ is analogous to the proof of [20, Prop. 7.24] and ‘(ii)’ was only given for completeness. The particular example can be found in the same reference.

We now consider the functional n-polyconvex hulls of a set. They are defined as follows.

Definition 4.55. Let K ⊆ R^d×D. Then we denote by P_n^fK all points that cannot be separated from K by a finite n-polyconvex function, i.e.

P_n^fK = {F ∈ R^d×D : f (F ) ≤ 0 for all f : R^d×D→ R n-polyconvex with f ≤ 0 on K}.

We call P_n^fK the functional n-polyconvex hull of K.

It is clear that we have the usual cascade of hulls, i.e. R^fK = P₁^fK ⊆ P₂^fK ⊆ . . . ⊆ P_d∧D−1^f K ⊆ P_d∧D^f K = P^fK, where P^fK and R^fK denote the usual functional polyconvex and rank-one convex hull of K. The following theorem asserts that the functional n-polyconvex hull is in general a stronger notion than the intersectional n-polyconvex hull (particularly for open sets).

Theorem 4.56. Let K ⊆ R^d×D. Then

P_n^∩K ⊆ P_n^fK.

The proof of this theorem is trivial since the functional n-polyconvex hulls are all closed, n-polyconvex sets. Thus, by definition P_n^∩K ⊆ P_n^fK. Furthermore, we suspect that the two hulls are inherently different in many cases. For example, we believe that there exist compact n-polyconvex sets (i.e. their intersectional n-polyconvex hull coincides with the set) which have a much larger functional n-polyconvex hull. These cases occur whenever there is a n-polyconvex set of finitely many points that form a T_k^n-pc configuration. Then at least the auxiliary points of the configuration must be included in the functional n-polyconvex hull. We conjecture the following.

Conjecture 4.57. For all 1 ≤ n < d ∧ D there exists a set K of finitely many points such that K is intersectionally n-polyconvex, but not functionally n-polyconvex.

The case n = 1 is dealt with by the well known T4 configuration in the plane. Con-sidering Example 4.45 we have proved that this is true for n = 2. We think that T_k^n-pc configurations also provide examples for n > 2.

Whenever a T_k^n-pc configuration exists in a set its functional n-polyconvex hull will be nontrivial. Thus, knowing whether such a configuration exists is helpful for computing

the n-polyconvex hulls of sets. For rank-one convexity and finite sets Kreiner et al. [33]

provide a way of testing for T_k configurations. Conversely, as proved in [59, Thm. 1], for a compact set in R^2×2 without any rank-one connections a nontrivial rank-one convex hull implies the existence of a T₄ configuration.

In general, for poly-, quasi- and rank-one convexity there exists a relationship between the functional semiconvex hull and the zero set of the respective semiconvex envelope of the distance function. The distance function dist_K to an arbitrary set K ⊆ R^d×D is defined as

distK(F ) = inf

F ∈Ke

||F − eF ||.

For quasi- and rank-one convexity the following theorem applies.

Theorem 4.58. Let K ⊆ R^d×D be a compact set. Then the functional ∗-convex hull of K is given by the zero set of the ∗-convex envelope of the distance function, i.e.

K^∗ = {F ∈ R^d×D : dist^∗_K(F ) = 0}, where ∗ ∈ {qc, rc}.

The case for quasiconvexity was proved by Zhang [62] and for rank-one convexity (in fact in greater generality for directional convexity) by Matouˇsek [39]. The case for polyconvexity is not as straightforward. Here the zero set depends on the power p of the p-distance function dist^p_K, whereas for quasi-, cf. [64, 63], and rank-one convexity this is not the case. ˇSilhav´y [53] shows that only for p ≥ d ∧ D does the same apply for the polyconvex hull of a compact set or when p = 1 one obtains the convex hull of the set instead. In fact, for any integer 1 ≤ p ≤ d ∧ D ˇSilhav´y defines the so-called s-polyconvex hull for the notions of s-polyconvexity, which was first defined in [12]. Note that s-polyconvexity is a concept that unifies standard convexity and polyconvexity in the sense that for s = 1 it is equivalent to convexity and for s = d ∧ D it is equivalent to polyconvexity and should not be confused with n-polyconvexity as defined here. A function f : R^d×D → R is s-polyconvex if there exists a convex function g : R^{τ (s)} → R such that f (F ) = g(adj₁F, . . . , adj_sF ), where τ (s) = Ps

i=1 d i

_D

i
. The functional s-polyconvex hull K^(s)-pc of a compact set K and the s-polyconvex envelope f^(s)-pc of a function f is defined as usual (the parenthesis (s)-pc is used to distinguish from the n-polyconvex case n-pc).

Theorem 4.59 (Thm. 1.1, [53]). Let K ⊆ R^d×D be compact. Further let s be an integer with 1 ≤ s ≤ d ∧ D. Then the s-polyconvex hull K^(s)-pc of K is given by the zero set of

the s-polyconvex envelope (dist^p_K)^(s)-pc of the p-distance function of K, i.e.

K^(s)-pc= {F ∈ R^d×D : (dist^p_K)^(s)-pc(F ) = 0}, for p ∈ [s, s + 1) or d ∧ D ≤ p < ∞ when s = d ∧ D.

Note that the case s = d ∧ D produces the polyconvex hull. Rephrasing the above results for poly- and rank-one convexity into the language of n-polyconvexity we have that the functional n-polyconvex hull K^n-pc is given by the zero set of the n-polyconvex envelope of distⁿ_K for n = d ∧ D or n = 1 respectively. Thus it seems reasonable to conjecture the following:

Conjecture 4.60. Let K ⊆ R^d×D be compact. Then the functional n-polyconvex hull K^n-pc of the set K is given by the zero set of n-polyconvex envelope of the n-th power of the distance function, i.e.

P_n^fK = {F ∈ R^d×D : P_n(distⁿ_K)(F ) = 0}.

We were not successful in proving the claim as we encountered the similar difficulties as in Section 4.2.4 for Conjecture 4.33. One such difficulty is that we do not have a global function g that is convex on T (F + V ) for all F ∈ R^d×D and V ⊆ R^d×D simple rank-n that represents f in the usual way f = g ◦ T on the whole of R^d×D. Having access to such a global representative is by no means a guarantee to success, but it is for instance among the essential ingredients of the proof by ˇSilhav´y for s-polyconvexity. On the other hand Matouˇsek’s proof for directional convexity (applicable for n = 1) does not carry over for n > 1. The main obstacle here is the nonlocality of n-polyconvexity for n > 1.

In almost all cases the respective relaxation of the distance function (or any other function) cannot be found by analytical means. Instead numerical approaches are necessary. In the case of rank-one convexity Dolzmann [22] proved the convergence of an iterative convexification along rank-one lines to the rank-one convex envelope of a function when the mesh size approaches zero. A necessary requirement to allow to work on a finite set of points is that the respective function has to agree with its rank-one convex envelope outside a fixed ball in the domain. We will return to a slight modification of the proposed algorithm in Section 5.3.3 after the considerations in Chapter 5.