The author refers to the study of the functional \[ {\mathcal J}_ H(X)= | \partial X|(\Omega)+ \int_ \Omega \phi_ X(x) H(x) dx, \] where \(\Omega\) is an open subset of \(\mathbb{R}^ n\) \((n\geq 2)\), \(H\in L'(\Omega)\), \(\phi_ X\) is the characteristic function of the measurable set \(X\subset \mathbb{R}^ n\) and \(|\partial X|(\Omega)\) is the perimeter of \(X\) in \(\Omega\). The second author [Arch. Ration. Mech. Anal. 55, 357-382 (1974; Zbl 0305.49047) and Rend. Semin. Mat. Univ. Padova 53(1975), 37-52 (1976; Zbl 0358.49019)] has shown that if \(E\) minimizes \({\mathcal J}_ H(X)\) with \(H\in L^ p(\Omega)\), \(p> n\) (\(E\) is said to have mean curvature \(H\)), then there exists an open subset \(\Omega_ 1\subset \Omega\) such that \(\Omega_ 1\cap \partial E\) is a hypersurface of class \(C^{1,\alpha}\), \(\alpha= {(p- n)\over 4p}\) and \(H_ s((\Omega- \Omega_ 1)\cap \partial E)= 0\), \(\forall s> n- 8\), where \(H_ s\) denotes \(s\)-dimensional Hausdorff measure in \(\mathbb{R}^ n\). The authors examine the sets of curvature \(H\in L^ n(\Omega)\) and in this process study a two-dimensional set with singular boundary. It is claimed that the singular boundary is the union of two spiral curves converging to the same point. Further, it is stated that the curvature of this line could be seen as the restriction of an \(L^ 2\) function as conjectured by \textit{E. De Giorgi} in a conference held at Trento in July 1992.