Isoperimetric Problems in Discrete Spaces (original) (raw)
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Discrete isoperimetric problems in spaces of constant curvature
Mathematika
The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with d + 2 vertices in Euclidean, spherical and hyperbolic d-space. In particular, we find the minimal volume d-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with d + 2 vertices with a given circumradius, and the hyperbolic polytopes with d + 2 vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any 1 ≤ k ≤ d, we investigate the properties of Euclidean simplices and polytopes with d + 2 vertices having a fixed inradius and a minimal volume of its k-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.
Some Isoperimetric Problems in Planes with Density
Journal of Geometric Analysis, 2010
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in R 2 . Finally, we give a list of related open questions.
The C onstrained Isoperimetric Problem
2011
Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio F R(S) = |S|/|∂S|. We consider different measures for subsets of R 2 , R 3 , Z 2 , Z 3 and describe the properties that must be satisfied for sets S that maximize the Følner ratio. We give explicit examples.
1997
We present here a description of all solutions of the isoperimetric problem in Hamming space of some special cardinalities. The number of these cardinalities equals 2 n?1 . Let B n denotes the vertex set of the n?dimensional unit cube with Hamming metric and A B n . Denote by S n k ( ) the sphere of radius k centered in 2 B n . We call a point 2 A the inner point of a set A if S n 1 ( ) A and the boundary point of A in the opposite case. Denote by P(A)?(A) the collection of all inner (boundary) points of A.
Parametrization of Isoperimetric-Type Problems in Convex Geometry
Аннотация. Choice of parametrization is considered for the isoperimetric-type extremal problems of optimal location of compact convex sets under many subsidiary constraints. Comparison is given between two parametrizations using support and surface area functions.
On the isoperimetric problem in Euclidean space with density
Calculus of Variations and Partial Differential Equations, 2007
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x| 2 ) by using symmetrization techniques. I f (V) = inf {P(Ω) : Ω is a smooth open set with vol(Ω) = V}. An isoperimetric region -or simply a minimizer-of volume V is an open set Ω such that vol(Ω) = V and P(Ω) = I f (V).
A Local-Global Principle for Vertex-Isoperimetric Problems
Dm, 1998
We consider the vertex-isoperimetric problem for cartesian powers of a graph G. A total order on the vertex set of G is called isoperimetric if the boundary of sets of a given size k is minimum for any initial segment of , and the ball around any initial segment is again an initial segment of . We prove a local-global principle with respect to the so-called simplicial order on G n (see Section 2 for the definition). Namely, we show that the simplicial order n is isoperimetric for each n ≥ 1 iff it is so for n = 1, 2. Some structural properties of graphs that admit simplicial isoperimetric orderings are presented. We also discuss new relations between the vertex-isoperimetric problems and Macaulay posets. * Partially supported by the Spanish Research Council under project TIC97-0963.