Langevin, Remi; Rosenberg, Harold Fenchel type theorem for submanifolds of \(S^ n\). (English) Zbl 0880.53047 Comment. Math. Helv. 71, No. 4, 594-616 (1996). The main result is: Let \(M\) be a compact hypersurface in the \(n\)-dimensional sphere, then \[ \sum^{n-1}_{i=0} c_iL_i (M)\geq \beta (M), \] where \(\beta (M)=\) sum of the Betti numbers of \(M\), \(L_i =i\)-length functional, defined integral-geometrically and localized through absolute curvature terms of \(M\), \(c_i=\) dimension constants. Further results of this nature deal with knotted curves and surfaces in spheres.Remark: Concerning the optimal Fenchel type theorem for curves in spheres cf. [E. Teufel, Manuscr. Math. 75, 43-48 (1992; Zbl 0754.53009)]. Reviewer: E.Teufel (Stuttgart) Cited in 1 ReviewCited in 1 Document MSC: 53C40 Global submanifolds 53C65 Integral geometry Keywords:Fenchel type theorem; submanifolds in spheres; knotted surfaces; \(i\)-length functional Citations:Zbl 0754.53009 PDFBibTeX XMLCite \textit{R. Langevin} and \textit{H. Rosenberg}, Comment. Math. Helv. 71, No. 4, 594--616 (1996; Zbl 0880.53047) Full Text: DOI EuDML