Iceberg‐type problems: Estimating Hidden parts of a continuum from the visible parts

Abstract

AbstractWe consider the complex plane \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}$\end{document} as a space filled by two different media, separated by the real axis \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}$\end{document}. We define \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}_+=\lbrace z: \,\Im \,z>0\rbrace$\end{document} to be the upper half‐plane. For a planar body E in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}$\end{document}, we discuss a problem of estimating characteristics of the “invisible” part, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E_-=E\setminus \mathbb {H}_+$\end{document}, from characteristics of the whole body E and its “visible” part, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E_+=E\cap \mathbb {H}_+$\end{document}. In this paper, we find the maximal draft of E as a function of the logarithmic capacity of E and the area of E+.