Hedgehog Theory

This book provides the very first comprehensive and self-contained introduction to hedgehog theory, which is born of the desire to visualize the formal differences of convex bodies. This extension of convex geometry has revealed unexpected depth and connections with many areas of mathematics, shedding new light on old problems such as the characterization of the 2-sphere conjectured by A.D. Alexandrov in the 1930s. The author is particularly keen to demonstrate the breadth and variety of applications of hedgehogs and their generalizations, in both geometry and analysis. Researchers in convex or differential geometry, as well as specialists in Monge-Ampère PDEs, will certainly find it a source of inspiration.

Chapter 1. Introduction.- Chapter 2. Background on classical real hedgehogs.- Chapter 3. Volumes and mixed volumes.- Chapter 4. Special convex bodies, hedgehogs or multihedgehog.- Chapter 5. The Minkowski problem for hedgehogs.- Chapter 6. Complex hedgehogs in Cn+1 or Pn+1 (C).- Chapter 7. Hedgehogs in non-Euclidean spaces.- Chapter 8. Marginally trapped hedgehogs.- Chapter 9. Focal of hedgehogs in Rn+1 and concurrent normals conjecture.- Chapter 10. Miscellaneous questions regarding hedgehogs.-Chapter 11. List of selected problems.

Yves Martinez-Maure is a French mathematician, a full professor at Sorbonne University and a member of the Institut de Mathématiques de Jussieu - Paris Rive Gauche since 2003. He earned his PhD from Paris 7 University in 1985, specializing in foliations. His subsequent research led to the development of hedgehog theory, a new subject which has become of particular interest in both convex and differential geometry.