Isoperimetric deficit and convex plane sets of maximum translative discrepancy
Let \(C\) be a simple closed plane curve of length \(\ell\) and enclosing an area \(a\). The isoperimetric deficit of \(C\) is \(\ell^ 2-4\pi a\). Then \(\ell^ 2\geq 4\pi a\) and \(\ell^ 2=4\pi a\) if and only if \(C\) is a circle \(S\) enclosing an area \(a\). The author asks what is the largest possible deviation of \(C\) from \(S\). He measures the deviation by a translative distance: the minimum distance between \(C\) and \(S\) when \(S\) is free to move in the plane. He answers the question using various distances.
scientific article; zbMATH DE number 64289