Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations

TL;DR

This paper introduces finite-dimensional variational principles connecting discrete optimal transport, Minkowski problems, and discrete Monge-Ampere equations, highlighting their geometric and computational relationships.

Contribution

It develops new variational frameworks for discrete optimal transport and related geometric problems, establishing links with power diagrams and computational geometry.

Findings

Unified variational principles for DOT, Minkowski, and DMAE

Connection between discrete Monge-Ampere and power diagrams

Foundations for computational algorithms in discrete geometry

Abstract

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computational geometry is established.