A mathematical assumption about the geometry of shapes that stood for 35 years has just been proven wrong.
This problem involves a specific rule of how the sizes and shapes of convex bodies behave when they are combined. Since 1988, mathematicians have suspected that a certain monotonicity principle always held true. This new study provides a counterexample that shows the rule can be broken. It overturns a long-standing conjecture in the field of affine geometry. This proof forces a complete re-evaluation of how we understand the fundamental math of shapes in higher dimensions.
On the monotonicity of affine quermassintegrals
arXiv · 2604.26828
Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn--Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov--Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts tha