Geometric inequalities | Analytic geometry | Calculus of variations | Multivariable calculus
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume , , where is a unit sphere. The equality holds only when is a sphere in . On a plane, i.e. when , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter". Specifically in , the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related Dido's problem asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found. The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. (Wikipedia).
Joe Neeman: Gaussian isoperimetry and related topics III
The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence
From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability
Joe Neeman: Gaussian isoperimetry and related topics II
The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence
From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability
The Isoperimetric - Larry Guth
Larry Guth University of Toronto; Member, School of Mathematics February 9, 2011 For more videos, visit http://video.ias.edu
From playlist Mathematics
Joe Neeman: Gaussian isoperimetry and related topics I
The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence
From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability
Isosceles & Equilateral Triangle Properties
I introduce 2 theorems about the properties of Isosceles and Equilateral Triangles. These theorems discuss how the base angles are congruent and that the bisector of the vertex is also a perpendicular bisector of the base. This video includes 2 proofs and 2 algebraic examples. EXAMPLES
From playlist Geometry
Higher order curvatures and isoperimetric inequalities - Yi Wang
Yi Wang Member, School of Mathematics October 1, 2014 More videos on http://video.ias.edu
From playlist Mathematics
Marston Morse - An Isoperimetric Concept for the Mass in General Relativity - Gerhard Huisken
Gerhard Huisken Max-Planck Institute for Gravitational Physics March 20, 2009 For more videos, visit http://video.ias.edu
From playlist Mathematics
Geometry and topology of isoperimetric and index one minimal surfaces - Celso dos Santos Viana
Short talks by postdoctoral members Topic: Geometry and topology of isoperimetric and index one minimal surfaces Speaker: Celso dos Santos Viana Affiliation: University College London; Member, School of Mathematics Date: September 26, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics
On minimizers and critical points for anisotropic isoperimetric problems - Robin Neumayer
Variational Methods in Geometry Seminar Topic: On minimizers and critical points for anisotropic isoperimetric problems Speaker: Robin Neumayer Affiliation: Member, School of Mathematics Date: February 19, 2019 For more video please visit http://video.ias.edu
From playlist Variational Methods in Geometry
Grigorios Paouris - Empirical Isoperimetric Inequalities outside convexity - IPAM at UCLA
Recorded 07 February 2022. Grigorios Paouris of Texas A&M University - College Station presents Empirical Isoperimetric Inequalities outside convexity at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: Several classical isoperimetric inequalities for convex s
From playlist Workshop: Calculus of Variations in Probability and Geometry
Eugenia Saorin-Gomez - Inner parallel bodies & the Isoperimetric Quotient
Recorded 10 February 2022. Eugenia Saorin-Gomez of the Universität Bremen presents "Inner parallel bodies & the Isoperimetric Quotient" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: The so-called Minkowski difference of convex bodies (compact and convex s
From playlist Workshop: Calculus of Variations in Probability and Geometry
Ilaria Fragalà: Some new inequalities for the Cheeger constant
Abstract: We discuss some new results for the Cheeger constant in dimension two, including: - a polygonal version of Faber-Krahn inequality; - a reverse isoperimetric inequality for convex bodies; - a Mahler-type inequality in the axisymmetric setting; - asymptotic behaviour of optimal par
From playlist Control Theory and Optimization
Steven Heilman - Variational Proofs of Isoperimetric Inequalities
Recorded 11 February 2022. Steven Heilman of the University of Southern California presents "Variational Proofs of Isoperimetric Inequalities" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: We will first survey some variational proofs of the Euclidean and
From playlist Workshop: Calculus of Variations in Probability and Geometry
How To Construct An Isosceles Triangle
Complete videos list: http://mathispower4u.yolasite.com/ This video will show how to construct an isosceles triangle with a compass and straight edge.
From playlist Triangles and Congruence
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
Colloquium MathAlp 2016 - Michel Ledoux
Isopérimétrie dans les espaces métriques mesurés Le problème isopérimétrique (à volume donné, minimiser la mesure de bord, et déterminer les ensembles extrémaux), remonte aux temps les plus anciens. Tout à la fois, il peut se formuler de façon générale dans un espace métrique mesuré, et d
From playlist Colloquiums MathAlp
How to use proportions for an isosceles triangle
👉 Learn how to solve with similar triangles. Two triangles are said to be similar if the corresponding angles are congruent (equal). Note that two triangles are similar does not imply that the length of the sides are equal but the sides are proportional. Knowledge of the length of the side
From playlist Similar Triangles