The Waist Inequality in Gromov’s Work

The waist inequality is a fundamental fact of Euclidean geometry. It’s also a difficult theorem it’s much harder to prove than it may look at first sight. In my opinion, the waist inequality is one of the most underappreciated theorems in geometry, and so I am excited to write about it. The waist inequality also connects with several other areas of mathematics. Gromov began writing about the waist inequality in the early 80’s, and he came back to it many times since then. When he started writing, the waist inequality could be proven as a corollory of deep work in geometric measure theory. Gromov gave several other proofs of the theorem, trying to get towards the bottom of this fundamental fact of geometry. He recognized and popularized the theorem, and gave a number of applications in geometry. More recently, he wrote several papers connecting the waist inequality to other areas of mathematics, such as combinatorics and topology. The isoperimetric inequality began as a theorem about Euclidean space. Later, people began to think about isoperimetric inequalities on other spaces, and they became a fundamental concept in geometry. Still later, people realized that many situations in different parts of mathematics are analogous to the isoperimetric inequality. Isoperimetric inequalities now play an important role in parts of group theory, graph theory, analysis, probability, computational complexity, and many other fields. In Gromov’s recent work, the waist inequality is beginning to play a similar role.