In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov–Fenchel inequalities. In particular, for $n = 2$ we obtain a Minkowski-type inequality and for $n = 3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.
"Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball." J. Differential Geom. 120 (2) 345 - 373, February 2022. https://doi.org/10.4310/jdg/1645207496