Related papers: An overview of complex ellipsoids

Abstract In this paper, we study affine bodies of revolution. This will allow us to prove that a convex body all whose orthogonal $n$-projections are affinely equivalent is an ellipsoid, provided $n\equiv 0,1, 2$ mod $4$, $n>1$ with the…

In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along…

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these…

The image and the inverse image of a polyhedron under a linear transformation are polyhedrons.

We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the…

We introduce the class of piecewise convex transformations, and study their complexity. We apply the results to the complexity of polygonal billiards on surfaces of constant curvature.

Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely…

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some…

We propose a simple derivation of an upper bound for the perimeter of an ellipse. The procedure, which relies on the use of elliptic integrals, consists in introducing, via inequalities and convexity properties, specific integrals which can…

Based on a novel discretization procedure which has recently been proposed and applied in the construction of a canonical discrete analogue of confocal coordinate systems, an explicit method of constructing discrete analogues of ellipsoids…

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They…

Gale transform is a simple but powerful tool in convex geometry. In particular, the use of Gale transform is the main argument in the classification of polytopes with few vertices. Many books and documents cover the definition of Gale…

Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate…

We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. These ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding…

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this…

We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In…

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…