Alexandrov--Fenchel inequalities for weighted curvature integrals in space forms

We establish two types of Alexandrov--Fenchel inequalities for the weighted curvature integrals of static convex domains in space forms. The first type consists of a complete family of inequalities involving three weighted curvature integrals, with the second Minkowski type inequality of Xia [Calc. Var. Partial Differential Equations 55 (2016), Art. 96] and the weighted almost Schur lemma of Li and Xia [J. Differential Geom. 113 (2019), pp. 493--518] as special cases. The second type consists of a complete family of inequalities involving two weighted curvature integrals, confirming a conjectured geometric inequality by Hu and Li [Trans. Amer. Math. Soc. 375 (2022), pp. 5587--5615]. The key novelty lies in observing that, via the orthogonal projection map, the weighted curvature integrals for domains in space forms admit a natural interpretation in terms of mixed volumes of Euclidean convex bodies, thus reducing the problem to the classical Alexandrov--Fenchel inequalities in convex geometry.