
Recent Posts
 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
Meta
Recent Comments
tong cheung yu on Toric perspective 1 Miodrag Mateljevic on Principle of subordination Lawrence G Mouille on Exponential maps of Lie g… tong cheung yu on On the existence of a metric c… Marco Barchiesi on Simple curves with a positive… tong cheung yu on Toric perspective 1 lamwk on Why is a² + b² ≥ 2ab ? Maurice OReilly on Spherical cosine law KKK on Sobolev and Isoperimetric… Anonymous on Sobolev and Isoperimetric… Categories
 Algebra
 Algebraic geometry
 Analysis
 Applied mathematics
 Calculus
 Combinatorics
 Complex analysis
 Differential equations
 Differential geometry
 Discrete Mathematics
 Dynamical system
 Fourier analysis
 Functional analysis
 General Relativity
 Geometry
 Group theory
 Inequalities
 Linear Algebra
 Miscellaneous
 Number Theory
 Operator Theory
 Optimization
 Potential theory
 Probability
 Set Theory
 Statistics
 Topology
 Uncategorized
Top Posts
 Complex analysis  Problem solving strategies.
 Martingale Theory III: Optional stopping theorem
 Martingale Theory II: Conditional expectation
 Sobolev and Isoperimetric Inequality
 Taylor expansion of metric
 Lie groups with biinvariant Riemannian metric
 Spherical cosine law
 AMGMHM Inequality: A Statistical Point of View
 Dual norm in R^n
 Sum of angle defects of polyhedrons
Archives
Category Archives: Inequalities
An inequality for functions on the plane
I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also … Continue reading
Posted in Analysis, Geometry, Inequalities
Leave a comment
Weighted isoperimetric inequalities in warped product manifolds
1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading
FaberKrahn inequality
I record a proof of the FaberKrahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading
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 CauchySchwarz inequality and the Lagrange identity
The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the crossterms part and the non crossterms part. The Lagrange identity implies the CauchySchwarz inequality in . And when , this can … Continue reading
Posted in Algebra, Group theory, Inequalities, Linear Algebra
Leave a comment
Weighted HsiungMinkowski 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
Posted in Calculus, General Relativity, Geometry, Inequalities
Leave a comment
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
Posted in Calculus, General Relativity, Geometry, Inequalities
Leave a comment