Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8900321 | Journal of Mathematical Analysis and Applications | 2018 | 16 Pages |
Abstract
Given a simple closed plane curve Î of length L enclosing a compact convex set K of area F, Hurwitz found an upper bound for the isoperimetric deficit, namely L2â4ÏFâ¤Ï|Fe|, where Fe is the algebraic area enclosed by the evolute of Î. In this note we improve this inequality finding strictly positive lower bounds for the deficit Ï|Fe|âÎ, where Î=L2â4ÏF. These bounds involve either the visual angle of Î or the pedal curve associated to K with respect to the Steiner point of K or the L2 distance between K and the Steiner disk of K. For compact convex sets of constant width Hurwitz's inequality can be improved to L2â4ÏFâ¤49Ï|Fe|. In this case we also get strictly positive lower bounds for the deficit 49Ï|Fe|âÎ. For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of 3, 4 or 5 cusps or the Minkowski sum of this kind of sets.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Julià CufÃ, Eduardo Gallego, Agustà Reventós,