Colorful Helly-type Theorems for the Volume of Intersections of Convex Bodies
Gábor Damásdi, Viktória Földvári, Márton Naszódi
2020ELTE Digital Institutional Repository (EDIT) (Eötvös Loránd University)14 citationsDOIOpen Access PDF
Abstract
We prove the following Helly-type result. Let C1,…,C3d be finite families of convex bodies in Rd. Assume that for any colorful selection of 2d sets, Cik∈Cik for each 1≤k≤2d with 1≤i1<…<i2d≤3d, the intersection ⋂k=12dCik is of volume at least 1. Then there is an 1≤i≤3d such that ⋂C∈CiC is of volume at least d−O(d2).
Topics & Concepts
Intersection (aeronautics)Regular polygonVolume (thermodynamics)CombinatoricsType (biology)MathematicsMixed volumeSelection (genetic algorithm)GeometryConvex bodyComputer scienceConvex hullGeologyGeographyPhysicsArtificial intelligenceCartographyPaleontologyQuantum mechanicsPoint processes and geometric inequalitiesComputational Geometry and Mesh GenerationLimits and Structures in Graph Theory