Discrete & Computational Geometry· 2026Q2
Random Shadows of Fixed Polytopes
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- Q2SCImago
- 2026year
Short summary
New upper bounds are proven for the expected number of vertices in a random 2D projection (shadow) of a fixed polytope, using parameters like geometric diameter, edge lengths, and slack.
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Key points
- Developed multiple upper bounds for the expected number of vertices in random 2D shadows of fixed polytopes.
- Bounds are expressed using parameters such as geometric diameter, edge lengths, slack, and subdeterminant values.
- Proved lower bounds for geometric diameter and edge length parameters, showing tightness for zonotopes.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool appearing in several applications through connections to linear programming and parametric optimization. We prove multiple upper bounds on the expected number of vertices of a random shadow of a fixed polytope. Our bounds are in terms of various parameters in the literature including geometric diameter and edge lengths, minimal and maximal slack, maximal coordinates for lattice polytopes, and maximum absolute values of subdeterminants. For the case of geometric diameter and edge lengths, we prove lower bounds and argue that our upper and lower bounds are both tight for zonotopes.
The authors' abstract, as published at the source. Discrete & Computational Geometry, 2026 · DOI ↗
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Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science