Results 1 
8 of
8
A CHARACTERIZATION OF HYPERBOLIC GEOMETRY AMONG HILBERT GEOMETRY
, 2007
"... In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point. We begin by recalling the Hilbert geometry of an open convex set. Let K be a nonempty bounded open convex set in R n, n ≥ 2. The Hilbert ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point. We begin by recalling the Hilbert geometry of an open convex set. Let K be a nonempty bounded open convex set in R n, n ≥ 2. The Hilbert distance dK on K is introduced by David Hilbert as follows. For any x ∈ K, let dK(x, x) = 0; for distinct points x, y in K, assume the line passing through x, y intersects the boundary ∂K at two points a, b such that the order of these four points on the line is a, x, y, b as in Figure 1. a x y b
Minimal Enclosing Hyperbolas of Line Sets
, 2006
"... We prove the following theorem: If H is a slim hyperbola that contains a closed set S of lines in the Euclidean plane, there exists exactly one hyperbola Hmin of minimal volume that contains S and is contained in H. The precise concepts of “slim”, the “volume of a hyperbola ” and “straight lines or ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
We prove the following theorem: If H is a slim hyperbola that contains a closed set S of lines in the Euclidean plane, there exists exactly one hyperbola Hmin of minimal volume that contains S and is contained in H. The precise concepts of “slim”, the “volume of a hyperbola ” and “straight lines or hyperbolas being contained in a hyperbola ” are defined in the text.
Sharpening Geometric Inequalities using Computable Symmetry Measures
, 2014
"... Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimen ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.
SETS OF UNIT VECTORS WITH SMALL SUBSET SUMS
"... ABSTRACT. We say that a family {xi i ∈ [m]} of vectors in a Banach space X satisfies the kcollapsing condition if ‖∑i∈I xi ‖ ≤ 1 for all kelement subsets I ⊆ {1,2,...,m}. Let C (k,d) denote the maximum cardinality of a kcollapsing family of unit vectors in a ddimensional Banach space, where th ..."
Abstract
 Add to MetaCart
(Show Context)
ABSTRACT. We say that a family {xi i ∈ [m]} of vectors in a Banach space X satisfies the kcollapsing condition if ‖∑i∈I xi ‖ ≤ 1 for all kelement subsets I ⊆ {1,2,...,m}. Let C (k,d) denote the maximum cardinality of a kcollapsing family of unit vectors in a ddimensional Banach space, where the maximum is taken over all spaces of dimension d. Similarly, let C B(k,d) denote the maximum cardinality if we require in addition that ∑ m i=1 xi = o. The case k = 2 was considered by Füredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finitedimensional Banach spaces. We show that C B(k,d) = max{k + 1,2d} for all k,d ≥ 2. The behaviour of C (k,d) is not as simple, and we derive various upper and lower bounds for different ranges of k and d. These include the exact values C (k,d) = max{k + 1,2d} in various cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the HajnalSzemerédi Theorem, the BrunnMinkowski inequality, lower bounds for the rank of a perturbation of the identity matrix.
MINIMAL AREA ELLIPSES IN THE HYPERBOLIC PLANE
"... Abstract. We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought among all ellipses with prescribed axes or center. In the general case, we present a sufficient and easily verifiable criterion on the enc ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought among all ellipses with prescribed axes or center. In the general case, we present a sufficient and easily verifiable criterion on the enclosed set that ensures uniqueness.