The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

Area of Science:

• Discrete Geometry
• Convex Geometry
• Number Theory

Background:

• Lattice polytopes are fundamental objects in discrete and convex geometry.
• Understanding their covering radius is crucial for various applications, including coding theory and sphere packing.
• Existing bounds often lack generality or are difficult to compute.

Purpose of the Study:

• To explore and establish upper bounds on the covering radius of non-hollow lattice polytopes.
• To investigate the relationship between covering radius and discrete geometric properties.
• To introduce and validate a discrete analog of Hadwiger's formula.

Main Methods:

• Conjecturing a general upper bound of d/2 for the covering radius in dimension d.
• Proving the conjecture for dimensions up to three.
• Establishing equivalence with the González-Merino and Schymura conjecture on covering minima.
• Introducing a novel concept of discrete surface area for lattice simplices.
• Proving a discrete analog of Hadwiger's formula in dimension two.

Main Results:

• A conjecture for a general upper bound of d/2 on the covering radius of non-hollow lattice polytopes.
• Proof of this conjecture up to dimension three.
• Equivalence shown between the main conjecture and the covering minimum of standard terminal simplices.
• Introduction and partial proof (dimension two) of a discrete analog of Hadwiger's formula.

Conclusions:

• The d/2 upper bound for the covering radius of non-hollow lattice polytopes is proven up to dimension three.
• The study provides strong evidence for a discrete analog of Hadwiger's formula, with implications for future research in discrete geometry.