Petty’s conjectured inequality for projection bodies

Determine whether Petty’s conjectured inequality concerning projection bodies of convex bodies in R^n holds; that is, prove or disprove the inequality formulated by Petty (1971) relating projection bodies within affine convex geometry.

Background

The paper reviews classical constructions in convex geometry, notably the projection body map Π and the difference body map D, and their characterization via equivariant valuation properties. In this context, the authors note that projection and difference bodies are central to several major results and inequalities.

Among these, Petty’s conjectured inequality (1971) remains unresolved. The authors reference it as a notable open problem in the classical setting, underscoring the enduring significance of projection bodies in affine isoperimetric inequalities. While this paper focuses on functional analogues and shows non-existence of a projection-body-type valuation in their functional framework, it does not resolve Petty’s conjecture.