Convex geometry

Convex geometry is the study of convex sets, convex bodies, and the properties that arise from convexity — the condition that the line segment joining any two points in a set remains entirely within the set. Though it originated in classical questions about volumes and symmetries, convex geometry has become indispensable in functional analysis, optimal transport, and the study of fully nonlinear PDEs, where convexity conditions determine whether solutions exist, whether they are unique, and whether they remain regular. The Brunn-Minkowski inequality, the isoperimetric inequality in convex form, and the theory of mixed volumes provide the structural backbone for geometric inequalities across mathematics.

The field is not merely a collection of results about convex objects. It is a framework for understanding when global geometric constraints — like convexity — force local analytic regularity. The Alexandrov-Fenchel inequality and the theory of Minkowski problems demonstrate that convex bodies encode curvature information in their support functions, and that this encoding is reversible: one can reconstruct a convex body from its curvature measure. This two-way passage between geometry and measure is the engine behind much of modern geometric analysis.

The assumption of convexity is often dismissed as a convenient restriction that makes theorems provable. This is backwards. Convexity is not a simplifying assumption — it is a structural property that nature selects. In optimization, convex landscapes have unique minima; in PDEs, convex domains preserve regularity; in probability, convex sets support concentration phenomena. The theorems of convex geometry are not true despite convexity but because of it, and the field's central task is to understand why convexity is the default geometry of well-posed problems.