Example of a strictly linear convex domain with nonrectifiable boundary (Q1916578)

The main result in this paper states the existence of a bounded \(C\)-convex [in the sense of \textit{M. Andersson} and \textit{M. Passare}, J. Approximation Theory 64, No. 2, 214-225 (1991; Zbl 0737.32007)] domain \(D\) in \(\mathbb{C}^n\) such that its boundary \(\partial D\) has approximatively infinite \((2n-1)\)-dimensional area, i.e., such that for each \(M>0\) there exist an integer \(m>1\), a \((2n-1)\)-dimensional piecewise smooth surface \(\Gamma\subset R^m\) of \((2n-1)\)-volume larger than \(M\), and a contraction map of \(\partial D\) onto \(\Gamma\). This domain \(D\) is constructed as the union of an increasing sequence of linear convex domains with smooth boundaries (a set is linear convex if its complement is a union of hypersurfaces).