Are circles isoperimetric in the plane with density er?

Rose-Hulman Undergraduate Mathematics Journal, Dec 2015

We prove that an isoperimetric region in R2 with density er must be convex and contain the origin, and provide numerical evidence that circles about the origin are isoperimetric, as predicted by the Log-Convex Density Conjecture.

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Are circles isoperimetric in the plane with density er?

RHIT Undergrad. Math. J. Rose-Hulman Undergraduate Mathematics Journal Miguel A. Fernandez Truman State University 0 1 2 Niralle Shah 0 1 2 0 Ping Ngai Chung University of Cambridge 1 Luis Sordo Vieira University of Kentucky , USA 2 Willams College Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation - RoseHulman Undergraduate Mathematics Journal Are circles isoperimetric in the plane with density er? Ping Ngai Chung a Niralee Shah c Miguel A. Fernandez b Luis Sordo Vieira d Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: http://www.rose-hulman.edu/mathjournal aUniversity of Cambridge bTruman State University cWilliams College dUniversity of Kentucky Are circles isoperimetric in the plane with density er? Ping Ngai Chung Miguel A. Fernandez Luis Sordo Vieira Niralee Shah Acknowledgements: We thank our advisor Frank Morgan for his patience and invaluable input. We also thank Diana Davis, Sean Howe and Michelle Lee for their help. For his helpful comments, we would like to thank our anonymous referee. For funding, we thank the National Science Foundation for grants to Prof. Morgan and to the Williams College \SMALL" Research Experience for Undergraduates, Williams College for additional funding, and the Mathematical Association of America for supporting our trip to speak at MathFest 2011. Introduction There has been a recent surge of interest in Riemannian manifolds with a positive \density" function that weights volume and area (see recent works by Morgan [M1-4]). In fact, Perelman [P, Section 1.1] used the concept of manifolds with density in his beautiful proof of the Poincare conjecture (see [M4] for an intuitive explanation). Although one might choose to weight area and perimeter di erently, the case where both the area and the perimeter are weighted by the same function is of most interest. A fundamental problem in manifolds with density is the isoperimetric problem, which seeks to enclose given area or volume with the least amount of perimeter. For example, it is well known that in the Euclidean plane, a circle encloses given area with least perimeter (for a historical treatment, see [B]). Due to its many applications in probability theory and in Perelman's proof of the Poincare conjecture, one of the most important examples is Rn with Gaussian density e r2: In 1975, C. Borell [B1] and V.N. Sudakov and B.S. Tsirel'son [ST] proved that half-spaces are perimeter minimizing for the Gaussian density. On the other hand, in Rn with density e+r2, C. Borell ([B2], see [Ro, Introduction and Theorem 5.2]) proved that balls about the origin are perimeter minimizing. Rosales et al. gave a conjecture on when balls about the origin are isoperimetric: Log-Convex Density Conjecture 1.1. [Ro, Conjecture 3.12] Consider Rn with a smooth radial density. If the log of the density is convex, then balls about the origin provide isoperimetric regions of any given volume. The borderline case of density er, whose log is linear and hence just barely convex, is a pivotal case and the subject of the following study. Although we are not able to prove that the isoperimetric solution is a circle about the origin, we obtain the following partial result: Theorem 3.19. In the plane with density er, an isoperimetric region is convex and contains the origin in its interior. The proof uses symmetrization, the Four-Vertex Theorem, and the equation for geodesic curvature. We conclude Section 3 with some numerical analysis. Relative to a given density e there is a generalization of curvature . Just as classically an isoperimetric curve (circle) has constant curvature, in the presence of a density an isoperimetric curve has constant generalized curvature. Corollary 3.23 provides a simple di erential equation for curves r( ) of constant generalized curvature : dr d = r s r 1 (r e rC)2 1 for some constant C ( 1 )e1=( 1 )= . Given , C, and initial position r(0), there are two solutions to the di erential equation: a nondegenerate solution that looks like one of the curves in Figures 1 and 2, and a degenerate solution that corresponds to a circle about the = 1:2, origin that does not have constant generalized curvature (Figure 3). When C reaches its minimum, the nondegenerate solution coincides with the degenerate solution and gives the unique constant generalized curvature curve, which is precisely a circle about the origin. To show that discs about the origin are isoperimetric, it su ces to show that a nondegenerate solution is not symmetric with respect to the line through the origin and a maximum point of r when C is strictly greater than its minimum, and therefore cannot be isoperimetric (Proposition 2.5). Thus the only remaining candidate for an isoperimetric curve is a circle about the origin, as desired. The structure of the paper is as follows: Section 2 provides basic de nitions and known results. Section 3 begins (...truncated)


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Ping Ngai Chung, Miguel A. Fernandez, Niralle Shah, Luis Sordo Vieira. Are circles isoperimetric in the plane with density er?, Rose-Hulman Undergraduate Mathematics Journal, 2015, pp. 12, Volume 16, Issue 1,