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Alexandrov-Fenchel type inequalities with convex weight in space forms
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DOI：10.1007/s10231-026-01663-7.png)
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Abstract
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In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family of geometric inequalities, as each convex, non-decreasing function yields a corresponding inequality, providing considerable flexibility. In particular, our results unify and extend a number of classical unweighted inequalities and their weighted extensions across different geometric settings. Finally, as an application of the weighted inequalities derived in our work, we establish a sharp upper bound for the first non-zero eigenvalue of a class of differential operators associated with k-convex hypersurfaces in R-n.
Keywords:
Weighted Alexandrov-Fenchel inequality
Space forms
Inverse curvature flow
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0.9
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75
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