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Isoperimetric inequalities and concave functions

Alane Susan Ferland, University of Massachusetts Amherst


In this thesis we prove a one parameter family of Bonnesen-style and Osserman-style discrete Sobolev inequalities which hold for a large class of concave functions. We will show that this class of concave functions include solutions of some well known second order differential equations. In particular, for the sine and cosine functions we will apply these discrete Sobolev inequalities to obtain uncountably many new Bonnesen-style and Osserman-style isoperimetric inequalities for the generalized star polygon Pn,m of type {[special characters omitted]}. We will also see that for certain classes of polygons P n we are able to give a complete solution for the existence of Bonnesen-style and Osserman-style isoperimetric inequalities. Our inequalities will be extended further to include the surfaces of constant curvature and prove global Bonnesen-style and Osserman-style inequalities for the generalized star polygon on the unit 2-sphere, hyperbolic plane and Euclidean plane. Finally, derived from the geometry of the surface, we develop Hyperbolic and Elliptic lengths and areas to prove additional interesting inequalities on the unit 2-sphere and hyperbolic plane.

Subject Area


Recommended Citation

Ferland, Alane Susan, "Isoperimetric inequalities and concave functions" (1999). Doctoral Dissertations Available from Proquest. AAI9932309.