Existence and uniqueness of solutions to Minkowski-type problems

The Minkowski problem is a highly significant problem in convex geometry and differential geometry, which concerns the construction of a convex body from a prescribed geometric measure. Mathematicians such as Minkowski, Aleksandrov, Nirenberg, Cheng-Yau, and Caffarelli have made major contributions to the existence, uniqueness, and regularity of solutions to the Minkowski problem. Over the past century, research on the Minkowski problem and its applications has greatly advanced the development of the Monge-Ampère equation. As the theory of the Minkowski problem has matured, it has gradually permeated other fields. Studies on related Minkowski problems have significantly promoted interdisciplinary integration between convex geometry and other fields. In this project, we seek to investigate Minkowski problems in various fields such as convex geometry, integral geometry, calculus of variations, and probability theory, by employing techniques from variational methods, continuity methods and degree theory in partial differential equations, as well as geometric flows. The research focuses on dual Minkowski-type problems in convex geometry, chord Minkowski-type problems in integral geometry, torsional Minkowski-type problems in the calculus of variations, and Gaussian Minkowski-type problems in probability theory. Through refining existing techniques and proposing novel approaches, the study aims not only to substantially address the challenges of existence and uniqueness for these Minkowski-type problems, but also to advance the theoretical framework of related Minkowski problems, thus expanding our previous works. More importantly, an in-depth investigation of these questions will provide new insights to solving longstanding open problems in their respective fields. The overarching goal of this project is to deepen the theoretical understanding of Minkowski problems, thereby fostering interdisciplinary integration across geometric analysis, partial differential equations, integral geometry and probability theory, and to strengthen the theoretical foundation provided by pure mathematics for applied fields such as physics and engineering.