Scholarlink

<p>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 <i>k</i>-convex hypersurfaces in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> </InlineEquation>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Alexandrov–Fenchel type inequalities with convex weight in space forms

  • 期刊论文

  • 发表日期：

    2026年2月7日

查看全文
  • Kwok-Kun Kwong,
  • Yong Wei

摘要

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 \(\mathbb {R}^n\) .