A cosine inequality in the hyperbolic geometry

A Strong Triangle Inequality in Hyperbolic Geometry

For a triangle in the hyperbolic plane, let α, β, γ denote the angles opposite the sides a, b, c, respectively. Also, let h be the height of the altitude to side c. Under the assumption that α, β, γ can be chosen uniformly in the interval (0, π) and it is given that α + β + γ < π, we show that the strong triangle inequality a + b > c + h holds approximately 79% of the time. To accomplish this, ...

A Matrix Hyperbolic Cosine Algorithm and Applications

Wigderson and Xiao presented an efficient derandomization of the matrix Chernoff bound using the method of pessimistic estimators [WX08]. Building on their construction, we present a derandomization of the matrix Bernstein inequality which can be viewed as generalization of Spencer’s hyperbolic cosine algorithm [Spe77]. We apply our construction to several problems by analyzing its computationa...

Hyperbolic Geometry

Problems in Hyperbolic Geometry

The purpose of this document is to list some outstanding questions in the subject of hyperbolic geometry (real, complex, quaternionic, and the Cayley plane), with the primary focus on the latter three. This list was inspired by the problem sessions held at the conference on Complex Hyperbolic Geometry in Luminy (CIRM) in July of 2003. The questions are roughly grouped by similarity. The name(s)...

Topics in Hyperbolic Geometry

There are two types of parallel lines in Hyperbolic Geometry. There are those who diverge from each other in both directions (type 1) and those that diverge in one direction but come arbitrarily close to each other in the other direction (type 2). The first type have a minimum positive distance at points on a common perpendicular. The second type have no minimum distance: the distance tends to ...