Neighborly Embedded Manifolds

Neighborly Embedded Manifolds

An embedding of an n-dimensional manifold M into R is called k-neighborly if, for every k points on the embedded manifold, there is a hyperplane H in R which supports the manifold precisely at these points. Micha A. Perles (Problems presented in Oberwolfach conference on “Convexity”, 1982) asked: What is the smallest dimension d(k,n) of the ambient space in which a k-neighborly n-dimensional ma...

Neighborly combinatorial 3-manifolds with 9 vertices

Cones Embedded in Hyperbolic Manifolds

We show that the existence of a maximal embedded tube in a hyperbolic n-manifold implies the existence of a certain conical region. One application is to establish a lower bound on the volume of the region outside the tube, thereby improving estimates on volume and estimates on lengths of geodesics in small volume hyperbolic 3-manifolds. We also provide new bounds on the injectivity radius and ...

Neighborly Cubical Polytopes

Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical convex d-polytope C d whose r-skeleton is combinatorially equivalent to that of the n-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂C d of a neighborly cubical polytope C n d maximizes the f -vector among all cubical (d− 1)-spheres with 2 vertices. While we sh...

Not Very Neighborly

North Carolina's swine industry is among the largest in the nation, second only to that of Iowa. By far, most of the hogs produced in North Carolina are raised in large industrial facilities with thousands of animals each. These operations have prompted concerns about noxious odors and potentially hazardous air emissions. Steve Wing, an associate professor of epidemiology in the School of Publi...