Volume of geometric objects plays an important role in applied and theoretical mathematics. This is particularly true in the relatively new branch of discrete geometry, where volume is often used to find new topics for research. Volumetric Discrete Geometry demonstrates the recent aspects of volume, introduces problems related to it, and presents methods to apply it to other geometric problems.
Part I of the text consists of survey chapters of selected topics on volume and is suitable for advanced undergraduate students. Part II has chapters of selected proofs of theorems stated in Part I and is oriented for graduate level students wishing to learn about the latest research on the topic. Chapters can be studied independently from each other.
I Selected Topics
Volumetric Properties of (m, d)-scribed Polytopes
Volume of the Convex Hull of a Pair of Convex Bodies
The Kneser-Poulsen conjecture revisited
Volumetric Bounds for Contact Numbers
More on Volumetric Properties of Separable Packings
II Selected Proofs
Proofs on Volume Inequalities for Convex Polytopes
Proofs on the Volume of the Convex Hull of a Pair of Convex Bodies
Proofs on the Kneser-Poulsen conjecture
Proofs on Volumetric Bounds for Contact Numbers
More Proofs on Volumetric Properties of Separable Packings
Open Problems: An Overview