On the nonlocality of n-polyconvexity

It is easy to define a nonnegative quadratic function f such that f (F ) = 0 if and only if F ∈ V . Again this f is polyconvex, but we would hope to modify it such that it remains 2-polyconvex, but not 3-polyconvex. To this end we proceed as above and define ε = inf{f (F ) : rank(F ) ≤ 2 and |F | = 1}. Then g defined through g(F ) = f (F ) − ε|F |² is quadratic and 2-polyconvex. However, at the present time we were unable to conclude whether g is 3-polyconvex or even polyconvex. The easiest path to successfully show that no α ∈ R^6×6exists such that

g(F ) − hα, adj₂F i ≥ 0 (4.64)

for all F ∈ R^4×4 would be to hope that for any α we can find F ∈ V such that (4.64) does not hold. Unfortunately the first five inequalities that α needs to satisfy by choosing each of the basis vectors above do not yield a contradiction like in R^3×3. It may however still be possible to show that a system of inequalities obtained by including more than just the five inequalities, but also other combinations of the five matrices does not allow a feasible solution. We did not pursue the matter further and it may well be that either it is not sufficient to search for counterexamples in V but instead in the whole of R^4×4 or that indeed g is actually 3-polyconvex or even polyconvex.

4.4.5. On the nonlocality of n-polyconvexity

Since the works of Kristensen [35, 34] it is known that both polyconvexity and quasicon-vexity are nonlocal concepts whereas rank-one conquasicon-vexity is equivalent to the positive definiteness of the second derivative of the function f at every point F ∈ R^d×D for all rank-one directions u ⊗ v with u ∈ R^d, v ∈ R^D, which is a local condition. To be more precise, in the case of either poly- and quasiconvexity, there exists a function

f : R^d×D→ R that is not poly- or quasiconvex, but agrees with a poly- or quasiconvex function on any ball of radius one around any point F ∈ R^d×D. This is true for d, D ≥ 2 in the case of polyconvexity and d ≥ 3, D ≥ 2 in the case of quasiconvexity. Here we extend these results to n-polyconvexity and we will prove that n-polyconvexity is nonlocal for d, D ≥ 2 and all n ≥ 2.

Theorem 4.66. Assume that d, D ≥ 2 and 2 ≤ n ≤ d ∧ D. Then there exists a smooth function f : R^d×D → R such that f is not n-polyconvex, but that its restriction to any ball B ⊆ R^d×D of radius one can be extended to a n-polyconvex function f_B : R^d×D→ R.

Remark 4.67. This is not a surprising result for two reasons. Firstly, n-polyconvexity requires the function f to have a convex representative for any simple rank-n subspace of R^d×D. For example, we can choose the space of the first n rows and columns being arbitrary and all others equal to zero, i.e. V = span{ei⊗ e_j : 1 ≤ i ≤ n, 1 ≤ j ≤ n} and there must exist g_V : R^{τ (d,D)}→ R such that f|V = g_V ◦ T on V . Defining a projection operator P : R^n×n → V with [P (F )]_ij = Fij if i, j ∈ {1, . . . , n} and [P (F )]ij = 0 otherwise, the condition f |V = gV ◦ T on V is equivalent to the function ef = f ◦ P being polyconvex. Thus, by carefully choosing the function f we can refer the question of nonlocality for n-polyconvexity back to simply polyconvexity. We will do this in detail in the following proof.

The second reason is that each representative g_V for each simple rank-n subspace V is defined on the set T (V ) and for all n > 1 the set T (V ) is a nonconvex. As Busemann et al. [16] remark, convexity on nonconvex sets is in general nonlocal and there is no reason to expect anything different in this particular case. On the other hand, when n = 1, T (V ) is a convex set already and thus convexity of gV is local. Therefore, rank-one convexity is local.

Proof. The proof is fairly straightforward and is based on the observation that any counterexample f2×2 to the locality of polyconvexity in the R^2×2 case also provides a counterexample f in R^d×D for d, D ≥ 2 by simply defining

f (F ) := f2×2

"

F₁₁ F₁₂ F₂₁ F₂₂

#!

.

With this definition it is known that f is polyconvex if and only if f_2×2is polyconvex.

Here we show that f is n-polyconvex for any 2 ≤ n ≤ d ∧ D if and only if f2x2 is polyconvex. It is easy to see that if f2×2 is polyconvex that then f is n-polyconvex for n ≤ d ∧ D (since f must be polyconvex). Thus it only remains to show that f being

2-polyconvex implies that f_2×2 must be polyconvex, which is sufficient for the statement to hold for n ≥ 2. Thus assume that f is 2-polyconvex. Then consider the simple rank-2 subspace V = span{ei⊗ e_j : i, j ∈ {1, 2}}, i.e. the space of all matrices

F =



















F₁₁ F₁₂ 0 . . . 0 F21 F22 0 . . . 0 0 0 0 . . . 0 ... ... ... ... 0 0 0 . . . 0



















∈ V ⊆ R^d×D.

Since f is 2-polyconvex there exists g_V : co(T (V )) → R such that f = gV ◦ T . Now define g : R⁵ → R such that g(X) = gV(XV) where XV ∈ R^{τ (d,D)} is the element that contains mostly zeros and only the possible five nonzero entries from X at their corresponding places in the bigger space. Then g inherits its convexity from g_V and we also obtain for F^2×2 ∈ R^2×2 and F ∈ V with Fij = F_ij^2×2 for i, j = 1, 2 that g(T (F^2×2)) = gV(T (F )) = f (F ) = f2×2(F^2×2) and so f2×2 is polyconvex. Therefore, by Kristensen’s result [35], n-polyconvexity is nonlocal since there exists a function f that is not n-polyconvex but agrees with a polyconvex function (and thus n-polyconvex function) on any ball of radius one.

5. A generalisation of abstract convexity

The previous chapter introduced the concept of n-polyconvexity and one of the main findings in the theory of n-polyconvexity was that any finite strong n-polyconvex function can be written as the pointwise supremum of n-polyaffine functions at the respective points. This is reminiscent of the concepts used in convex analysis for convex functions and has been made abstract in the theory of abstract convexity to include cases like polyconvexity as the supremum of polyaffines (known as quasiaffines in the current literature) as well as the relevant notions with respect to sets. The field of abstract convexity has many contributors and a comprehensive list of references can be found in the monographs [51, 55, 46]. The main point to be made here is that none of the contributions are capable of including the case of rank-one convexity (and n-polyconvexity for n < d ∧ D) into the abstract convexity framework. However, with the work of the previous chapter we can see that it is possible to generalise the concepts of abstract convexity to allow for such an inclusion. Abstract convexity rests on the observation that convex objects like functions or sets can be characterised by a simpler class of convex objects, e.g. affine functions or halfspaces, and that this characterisation does not in particular depend on the structure of the underlying involved spaces. Therefore, the methods of convex analysis can be generalised to include cases like polyconvexity, the essence of which is captured in abstract convexity theory. The way forward with respect to n-polyconvex functions is to allow the above elementary functions, in our case n-polyaffine functions, to depend on the point at which they form the supremum. Hence, this chapter will be devoted to this generalisation of abstract convexity.