Category Archives: Inequalities

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

The Cauchy-Schwarz inequality and the Lagrange identity

The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part. The Lagrange identity implies the Cauchy-Schwarz inequality in . And when , this can … Continue reading

Posted in Algebra, Group theory, Inequalities, Linear Algebra | Leave a comment

Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

1. Motivation and Main Results A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in are round hyperspheres. The embeddedness assumption is essential. For instance, admits immersed tori with constant mean … Continue reading

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A functional inequality on the boundary of static manifolds

1. introduction and statement of results The research in this article is largely motivated by the following result concerning a functional inequality on the boundary of bounded domains in the Euclidean space , proved in [MTX] Corollary 3.1. Theorem 1 … Continue reading

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The Brunn-Minkowski inequality and the isoperimetric inequality

In this short note, I will prove the Brunn-Minkowski inequality and use it to derive the isoperimetric inequailty. Of course, the theory is quite well-known and I am writing it for my own benefit. Let , we define the sum … Continue reading

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Some simple Ros type inequalities

In this short note, we prove some simple sharp geometric inequalities which are similar to Ros’s inequality. In [Ros], Ros proved that if is a smooth Riemannian manifold with boundary such that has positive normalized mean curvature and has non-negative … Continue reading

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Some more volume comparison results

Let be a Riemannian manifold. We prove two versions of volume comparison results. The first one compares the volume of a geodesic ball with a geodesic ball of the same radius in another Riemannian manifold (not necessarily a space form), … Continue reading

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