This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.

Longest minimal length partitions

While two-dimensional triangles are always subdivision invariant, the same does not always hold for their three-dimensional counterparts. We consider several interesting properties of those three-dimensional tetrahedra which are subdivision invariant and offer them a new classification. Moreover, we study the optimization of these tetrahedra, arguing that the second Sommerville tetrahedra are the closest to being regular and are optimal by many measures. Anisotropic subdivision invariant tetrahedra with high aspect ratios are characterized. Potential implications and applications of our findings are also discussed.

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Optimization of Subdivision Invariant Tetrahedra

We prove the theorem mentioned in the title for ℝn where n ≧ 3. The case of the simplex was known previously. Also the case n = 2 was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic n-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas and some partial results about sufficient conditions for the existence of (convex) tetrahedra.

The infimum of the volumes of convex polytopes of any given facet areas is 0

Ueber Maximum und Minimum bei den Figuren in der Ebene, auf der Kugelfläche und im Räume überhaupt.

This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points. 

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal "orientation-preserving" tetrahedral tile (n = 4), and we give a nice new proof for the optimal 5-hedron (a triangular prism).

/pdf/surface-area-minimizing-n-hedral-tiles-2qmu6w35hj.pdf

Surface-area-minimizing n-hedral Tiles

The Convex Body Isoperimetric Conjecture states that the least perimeter needed to enclose a volume within a ball is greater than the least perimeter needed to enclose the same volume within any other convex body of the same volume in Rn. We focus on the conjecture in the plane and prove a new sharp lower bound for the isoperimetric profile of the disk in this case. We prove the conjecture in the case of regular polygons and show that in a general planar convex body, the conjecture holds for small areas. Acknowledgements: This paper is the work of the Williams College NSF “SMALL” 2013, 2014, and 2015 Geometry Groups. We thank our advisor Professor Frank Morgan for his support. We would like to thank the National Science Foundation, Williams College, and the MAA for supporting the “SMALL” REU and our travels to MathFest 2013, MathFest 2014, and MathFest 2015. Page 10 RHIT Undergrad. Math. J., Vol. 18, No. 2

/pdf/the-convex-body-isoperimetric-conjecture-in-the-plane-3535x2oz7v.pdf

The Convex Body Isoperimetric Conjecture in the Plane

We study the isoperimetric problem in the plane with weighting or density e−1/r. The isoperimetric problem seeks to enclose prescribed weighted area with minimum weighted perimeter. For density e−1/r, isoperimetric curves are conjectured to pass through the origin. We provide numerical and theoretical evidence that such curves have an angle at the origin approaching 1 radian from above as area approaches zero and provide further estimates. Acknowledgements: This paper is the work of the SMALL 2013 Geometry Group, an undergraduate mathematics research group at Williams College advised by Frank Morgan. We thank Professor Morgan for his support and advice. For funding, we thank the National Science Foundation for grants to the Williams College SMALL Research Experience for Undergraduates, Williams College for additional funding, and the Mathematical Association of America, the University of Texas at Austin, University of Pennsylvania, and Georgetown University for supporting our trip to talk at MathFest 2013 and the 2014 Joint Mathematics Meetings. RHIT Undergrad. Math. J., Vol. 15, No. 2 Page 101

/pdf/isoperimetry-in-the-plane-with-density-e-1-r-1wyi1oato3.pdf

Byron Perpetua

Papers

Surface-area-minimizing n-hedral Tiles

The Convex Body Isoperimetric Conjecture in the Plane

Isoperimetry in the Plane with Density e-1/r