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Topology/Geometry Seminar
Friday, April 9, 2009
1:30 pm,
114 McAllister
Robert Foote
Wabash College
Planimeters and Isoperimetric Inequalities on Constant Curvature Surfaces.
Abstract: The well-known isoperimetric inequality states that $4\pi A \le L^2$, where $A$ is the area of a region in the Euclidean plane and $L$ is the length of its boundary. The corresponding inequality for regions on the sphere or in the hyperbolic plane is $4\pi A - kA^2 \le L^2$, where $k$ is the curvature of the surface.
A planimeter is a simple mechanical instrument used to determine the area of a planar region by tracing around its boundary. I will show how one works, including on the sphere and hyperbolic plane, and use the ideas involved to give a novel proof of some stronger Bonnesen isoperimetric inequalities on these surfaces.
We meet weekly on Friday afternoons at Penn State Altoona (and sometimes at Penn State University Park). If you are interested in giving a talk in our seminar please contact one of the coordinators for the Spring 2009 semester: Wojtek Dorabiala and Aissa Wade.
