A Set of Questions in Combinatorial and Metric Geometry

TL;DR

This paper introduces several open problems in combinatorial and metric geometry, exploring topics like fair partitions, polyhedron invariants, tiling, and extremal convex regions, providing partial insights and highlighting future research directions.

Contribution

It presents new questions and partial solutions in combinatorial and metric geometry, expanding understanding of geometric partitions, invariants, tiling, and extremal convex shapes.

Findings

Partial solutions to the convex fair partition conjecture

Identification of invariants among polyhedra from the same face set

Insights into tiling rectangles and extremal convex regions

Abstract

We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling rectangular tiles to form larger rectangles and (4) on convex regions which maximize and minimize the diameter for specified area and perimeter. For each question, we discuss partial solutions and indicate aspects that to our knowledge, await exploration.