On the length of simple closed quasigeodesics on convex surfaces (Q2506477)

Given a simply closed quasigeodesic \(O\) on a general convex surface \(S\), the lower and upper estimates \[ A_1 \leq ld_1 \leq 2A_1 \] for its length \(l\) in terms of the area \(A_1\) of the bounded domain \(S_1\) and the maximal radius \(d_1\) of inscribed balls (in the sense of the interior metric in \(S_1\)) are proved. Note that the nonnegative rotation property only from the side of the domain \(S_1\) in every point of \(O\) is used. From this viewpoint the result can not be considered as new. The stronger estimation from below for the value \(ld_1\) including the total positive curvature \(\omega_+\) in more general situation was proved in \textit{J. D. Burago} and \textit{V. A. Zalgaller} [Proc. Steklov Inst. Math. 76(1965), 100--108 (1967); translation from Tr. Mat. Inst. Steklov 76, 81--87 (1967; Zbl 0167.50802)]. The right hand inequality for smooth surfaces \(S\) is the simplest case of a theorem in Chapter 6, \S 5 of the book [\textit{J. D. Burago} and \textit{V. A. Zalgaller}, Geometric inequalities. Akad. Nauk SSSR, Leningr. Otd., Mat. Inst. Im. V. A. Steklova (1980; Zbl 0436.52009)]. The novelty is the formulation of the mentioned inequalities in the context of quasigeodesics. The conditions under which equality holds are indicated.