On relative isoperimetric inequalities in the plane. (English) Zbl 0674.49030

An isoperimetric inequality relative to an open bounded subset G of \({\mathbb{R}}^ n\) is said to hold if two positive constants and Q exist such that \[ (1)\quad P(E;G)\quad Q\geq [\min \{meas E,\quad meas G\setminus E\}]^{\alpha} \] for every measurable subset E of G. Here “meas” is Lebesgue measure and P(E;G) is the perimeter of E relative to G. The smallest number Q for which (1) holds is called the isoperimetric constant relative to G and will be denoted by Q(\(\alpha\) ;G). In our paper we consider the case \(n=2\). We show that, if \(G\subset {\mathbb{R}}^ 2\) is convex, then there exist subsets of G satisfying the equality in (1) with \(Q=Q(\alpha;G)\) and we give a characterization of such subsets. The isoperimetric constants relative to triangles, regular polygons and convex sets with a center of symmetry are computed. Finally, we get sharp estimates for Q(\(\alpha\) ;G) as G belongs to particular classes of convex sets.
Reviewer: A.Cianchi


49Q05 Minimal surfaces and optimization
49Q15 Geometric measure and integration theory, integral and normal currents in optimization