Geometric inequalities in normed spaces
Let \(x\) be a \(T\)-periodic solution of \(x'= f(x)\), where \(f: E\to E\) is Lipschitz continuous with constant \(L\) and \(E\) is a Banach space. In [the first author, \textit{D. Fisher} and the second author, Proc. Am. Math. Soc. 98, 376-378 (1986; Zbl 0632.34048)] it is shown that \(TL\geq 6\); this inequality is optimal [cf. the first author, \textit{D. Fisher} and the second author, Am. Math. Mon. 96, No. 1, 5-17 (1989; Zbl 0672.34034)]. The authors of the paper under review apply ideas and methods of the above mentioned works to solve some problems of geometrical nature. They obtain new proofs and/or new results which generalize the existing ones. Among other things, they extend the classical result of \textit{W. Fenchel} [Bull. Am. Math. Soc. 57, 44-54 (1951; Zbl 0042.400)] to any finite or infinite dimensional Hilbert space, they prove inequalities on sums of distances between finitely many vectors in normed spaces [special cases yield results from \textit{G. D. Chakerian} and \textit{M. S. Klamkin}, Am. Math. Mon. 80, 1009-1017 (1973; Zbl 0276.52005)], they improve an inequality concerning convex sets which was established by \textit{A. D. Andrew} and \textit{M. A. Ghandehari} [Congr. Numerantium 50, 31-35 (1985; Zbl 0592.52006)].
scientific article; zbMATH DE number 97419
scientific article