Area minimization of special polygons (Q2300111)

The paper under review concerns the following statement on a particular area minimization problem for convex polygons.\par Theorem (Hajós lemma). Fix a circle \(\gamma\) inside a concentric unit circle \(\Gamma\). Among all convex polygons which contain the smaller circle \(\gamma\) and have no vertices inside the unit circle \(\Gamma\), the polygon which is inscribed in \(\Gamma\) so that all its sides, with the exception of at most one side, is tangent to \(\gamma\) is of minimum area. The authors extend Hajós lemma to the case of non-concentric circles as follows. Theorem. Fix a small circle \(\gamma\) inside a unit circle \(\Gamma\) and assume that \(\gamma\) is non-concentric to \(\Gamma\). Consider all polygons which are inscribed in the unit circle \(\Gamma\) and contain the smaller circle \(\gamma\). Then the smallest area polygon has all its sides, with the exception of at most one side, tangent to \(\gamma\). Moreover, Hajós lemma is extended to the case of specific planar convex sets called disc polygons. By definition, an \(R\)-disc polygon is the intersection of finitely many congruent discs of radius \(R\). The boundary of a disc polygon consists of circular arcs called sides, the common points of adjacent sides are called vertices. An \(R\)-disc polygon with \(R>1\) is said to be inscribed in a unit circle if all its vertices lie on this unit circle. Then the following version of Hajós lemma is proved. Theorem. Let \(0<r<1\) and \(R>1\). Fix a circle \(\gamma\) of radius \(r\) inside of a concentric unit circle \(\Gamma\). Among all \(R\)-disc polygons which contain the circle \(\gamma\) and have no vertices inside the unit circle \(\Gamma\), the \(R\)-disc polygon which is inscribed in \(\Gamma\) so that all its sides, with the exception of at most one side, are tangent to \(\gamma\) is of minimum area.