Mathematics

We study six pedal-like curves associated with the ellipse which are area-invariant for pedal points lying on one of two shapes: (i) a circle concentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is a corollary to properties of the Curvature Centroid (Krümmungs-Schwerpunkt) of a curve, proved by Steiner in 1825. For case (ii) we prove area invariance algebraically. Explicit expressions for all invariant areas are also provided.

We present a characterization of minimal cones of class C 2 and C 1 in the first Heisenberg group H 1 , with an additional set of examples of minimal cones that are not of class C 1 .

We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.

We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the dimension and the volume of the manifold.

We generalize the concept of the Voronoi path of a line to more general shapes and compute the distortion constant, which describes how it changes volume on average. Although initially asked for a Poisson point process, the distortion is a characteristic property of the space rather than the point process. In other words, the constant ratio of the perimeter of a circle and its pixelation - and the analogous ratios for spheres in three and higher dimensions - hold for all smoothly embedded shapes on average.

We show that the Banach-Mazur distance between the parallelogram and the affine-regular hexagon is 3 2 and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just 3 2 . A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach-Mazur distance of any planar centrally-symmetric bodies is at most 3 2 . Analogously, we deal with the Banach-Mazur distances between the parallelogram and the remaining affine-regular even-gons.

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. This is inspired by results of Herron-Meyer and Rohde for quasi-arcs. Second, we show that a quasiconformal tree bi-Lipschitz embeds in a Euclidean space if and only if its set of leaves admits such an embedding. In particular, all quasi-arcs bi-Lipschitz embed into some Euclidean space.

Two decades ago, Zauner conjectured that for every dimension d , there exists an equiangular tight frame consisting of d 2 vectors in C d . Most progress to date explicitly constructs the promised frame in various dimensions, and it now appears that a constructive proof of Zauner's conjecture may require progress on the Stark conjectures. In this paper, we propose an alternative approach involving biangular Gabor frames that may eventually lead to an unconditional non-constructive proof of Zauner's conjecture.

We prove some recent experimental observations of D. Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.

We show that, for each 1≤p<2 , there exists a wild involution S 3 → S 3 in the Sobolev class W 1,p ( S 3 , S 3 ) .