A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translation-invariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one, the Hodge–Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge–Riemann relations for the convolution directly imply the Aleksandrov–Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov–Fenchel inequality for lower-dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by [S. Alesker, Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies, preprint (2020), arXiv:2010.01859].

估值的霍奇-黎曼关系的一个版本最近由 [J. Kotrbatý，关于平移不变估值的霍奇-黎曼关系，预印本 (2020)，arXiv:2009.00310]。那里考虑的 Lefschetz 算子要么是乘积，要么是与几个欧几里得球的混合体积的卷积。在这里，我们证明了在（共）度一中，如果球被几个不同的（中心对称）凸体替换，这些凸体具有平滑的边界和正高斯曲率，霍奇-黎曼关系仍然存在。虽然卷积的这些混合 Hodge-Riemann 关系直接暗示 Aleksandrov-Fenchel 不等式，但它们为乘积的对偶运算产生了一个新的不等式。这种新的不等式强化了 Aleksandrov-Fenchel 不等式对低维凸体的经典结果，并概括了 [S. Alesker，Kotrbatý 关于凸体估值和几何不等式的定理，预印本 (2020)，arXiv:2010.01859]。