By Achim Bachem (auth.), Prof. Dr. Peter M. Gruber, Prof. Dr. Jörg M. Wills (eds.)
This selection of surveys is composed partly of extensions of papers awarded on the meetings on convexity on the Technische Universitat Wien (July 1981) and on the Universitat Siegen (July 1982) and partly of articles written on the invitation of the editors. This quantity including the sooner quantity «Contributions to Geometry» edited by means of Tolke and Wills and released by means of Birkhauser in 1979 should still provide a reasonably strong account of some of the extra vital elements of convexity and its functions. along with being an up to the moment reference paintings this quantity can be utilized as a complicated treatise on convexity and comparable fields. We basically desire that it'll motivate destiny study. Fenchel, in his paper, offers an ancient account of convexity exhibiting many very important yet now not so popular aspects. The articles of Papini and Phelps relate convexity to difficulties of practical research on nearest issues, nonexpansive maps and the extremal constitution of convex units. A bridge to mathematical physics within the feel of Polya and Szego is equipped via the survey of Bandle on isoperimetric inequalities, and Bachem's paper illustrates the significance of convexity for optimization. The contribution of Coxeter offers with a classical subject in geometry, the strains at the cubic floor while Leichtweiss exhibits the shut connections among convexity and differential geometry. The exhaustive survey of Chalk on aspect lattices is said to algebraic quantity idea. a subject very important for purposes in biology, geology etc.
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Similar statements were proved in [Ber-Me] for surfaces in higher dimensions. Hersch [He1] showed that for convex domains with inradius Po the first eigenvalue of the homogeneous membrane satisfies A1 ~ C;J2, where the right-hand side is the limiting value for long thin rectangles. Other nonisoperimetric results relating A1 and Po are outlined in . Isoperimetric inequalities 43 Because of the variational characterization of the eigenvalues upper bounds are much easier to construct and there are numerous isoperimetric inequalities providing such bounds.
Math. (Szeged) 20 (1959),33-35. : On the principal frequency of a membrane, Pac. J. Math. 8 (1958),285-293. : The Isoperimetric Inequality, Bull. Amer. Math. Soc. 84 (1978),1182-1238. : Isoperimetric inequalities and eigenvalues of the Laplacian, Proceedings of [02J the International Congress of Mathematicians, Helsinki (1978). : Bonnesen-Style Isoperimetric Inequalities, Amer. Math. Monthly, 86 (1979), [03J 1-29. : Some isoperimetric inequalities in the torsion problem for multiply connected regions, Studies in Mathematical Analysis and Related Topics: Essays in honor of G.
For more recent results consult also [Ta, Ba5,7, Sc-Sp]. Consider now problem (14) in a simply connected domain D on a surface surface S Asu +1= 0 in D c ~2, U = 0 on aD. If the Gaussian curvature is bounded by Ko and if KoAiD) < 411:, then [Ba5] 1 I 411: umax :s;; Ko og 411: - KoA,,(D)· Equality holds for geodesic circles on surfaces of constant Gaussian curvature Ko. More on this problem is found in [Bal, 2, 5, 7]. In particular problems with mixed boundary conditions are considered. 46 C. Bandle Finally, we remark that we have not tried to give a complete up-to-date picture of the complex and diversified body of isoperimetric inequalities, but we have rather tried to collect the material related with the two-dimensional surfaces.