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 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
 A curious identity on the median triangle
 27 lines on a smooth cubic surface
 Weighted HsiungMinkowski formulas and rigidity of umbilic hypersurfaces
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 Complex analysis  Problem solving strategies.
 Lie groups with biinvariant Riemannian metric
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 Martingale Theory II: Conditional expectation
 Some integral formulas for hypersurface in Euclidean space 2
 Some integral formulas for hypersurface in Euclidean space
 An inequality for functions on the plane
 Surjectivity of Gauss map and its degree
 Weighted isoperimetric inequalities in warped product manifolds
 The discrete GaussBonnet theorem
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Author Archives: KKK
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
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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
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A remark on the divergence theorem
The divergence theorem states that for a compact domain in with piecewise smooth boundary , then for a smooth vector field on , we have where is the unit outward normal and is the divergence of . In most textbooks, … Continue reading
Posted in Calculus
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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
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On the existence of a metric compatible with a given connection
Question: Suppose we are given a torsionfree (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its LeviCivita connection is ? If so, is it unique if we … Continue reading
Posted in Differential equations, Geometry
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