Conditions for equality of hulls in the calculus of variations (Q1587319)

Let \(M^{m\times n}\) be the space of all real \(m\times n\)-matrices with the norm \(|x|^2=\text{tr}(x^Tx)\). Let \(K\subseteq M^{m\times n}\) be compact, \(K^c\) be the (closed) convex hull of \(K\), \(K^qc\) be the (closed) quasiconvex hull of \(K\), and \(K^pc\) be the (closed) polyconvex hull of \(K\) defined, respectively, as \[ K^qc=\{x : f(x)\leq\sup_{y\in K}f(y)\text{ for all }f\colon M^{m\times n}\to{\mathbf R}\text{ quasiconvex}\}, \] \[ K^pc=\{x : f(x)\leq\sup_{y\in K}f(y)\text{ for all }f\colon M^{m\times n}\to{\mathbf R}\text{ polyconvex}\}. \] Moreover, let \(L_0(K)=K\), and \[ L_{i+1}(K)=\{x=\lambda a+(1-\lambda)b : \lambda\in[0,1],\;a,b\in L_i(K),\text{rank}(a-b)\leq 1\}. \] The paper deals with the properties of \(K^qc\) and \(K^pc\), by means of an approach emphasizing the underlying geometry. By simplifying some results by K. Zhang, the authors prove, in particular, that \(K^pc=K^c\) implies \(L_{\dim (K)}=K^ c\) if and only if \(\min{m,n}\leq 2\), where \(\dim (K)\) is the affine dimension of \(K\). In this way, they answer a question raised in [\textit{K. M. Zhang}, Calc. Var. Partial Differ Equ. 6, No. 2, 143-160 (1998; Zbl 0896.49005)]. Moreover, they investigate the properties of the distance function \(d_{K,p}(x)=\inf\{|x-y|^p : y\in K\}\), when \(p\in[1,+\infty]\). In particular, they prove that \(d_{K,p}\) cannot be used as an indicator of whether a given set is quasiconvex or not.