On the conjectured capillary Blaschke-Santaló inequality

Abstract

We prove that the conjectured capillary Blaschke--Santaló inequality holds for any unconditional, strictly convex capillary hypersurface when $\theta \in \left(0,\frac{\pi }{2}\right)$. Moreover, for $\theta \in \left(\frac{\pi }{2},\pi \right)$, we show that the capillary volume product has no finite upper bound.