Some integral formulas on the unit sphere (Q2782737)

Let \(K\subset S^2\) be a spherically convex set, and let \(B_\rho(C)\) be the parallel set at distance \(\rho/2\) (parallel strip) of a great circle \(C\) of the two-dimensional unit sphere \(S^2\). Further, let \(\overline {u}(C)\), \(\overline{f}(C)\) denote the perimeter and the area of the convex hull of \(K\cap B_\rho(C)\). Then the authors derive estimates, in terms of \(\rho\) and the perimeter and area of \(K\), for the integrals of \(\overline{u}\) and \(\overline{u}^2-\overline{f}^2\) with respect to the rotation invariant measure on the space of great circles of \(S^2\). The quality of these estimates is not discussed. For small values of \(\rho>0\), some consequences of these integral estimates (Corollary 1 and Theorem 3) are trivially satisfied. NEWLINENEWLINENEWLINE[Remark: It seems that reference [3] of the paper, which is cited repeatedly, has appeared under a different title; cf. \textit{Y. D. Chai} and \textit{S. Y. Lee}, A Crofton style formula and its application on the unit sphere \(S^2\), Bull. Korean Math. Soc. 33, No. 4, 537-544 (1996; Zbl 0869.53049)].