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Triangle center
In geometry, the isoperimetric point is a triangle center — a special point associated with a plane triangle. The term was originally introduced by G.R
Isoperimetric_point
Triangle center
the isoperimetric point possesses the equal detour property as well. The equal detour point, isoperimetric point, the incenter and the Gergonne point of
Equal_detour_point
Geometric inequality applicable to any closed curve
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the
Isoperimetric_inequality
Geometric concept
detour point; otherwise the equal detour point is unique. When the outer Soddy circle has negative curvature, its center is the isoperimetric point of the
Soddy_circles_of_a_triangle
Equation for radii of tangent circles
through the third], pp. 128–144 Veldkamp, G. R. (1985), "The Isoperimetric Point and the Point(s) of Equal Detour in a Triangle", The American Mathematical
Descartes'_theorem
In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio L2/A, where L is the length of the curve and
Isoperimetric_ratio
List of points considered center of a triangle
Each point in the list is identified by an index number of the form X(n) —for example, X(1) is the incenter. The information recorded about each point includes
Encyclopedia of Triangle Centers
Encyclopedia_of_Triangle_Centers
Point in a triangle that can be seen as its middle under some criteria
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example
Triangle_center
Isodynamic point Isogonal conjugate Isoperimetric point Isosceles triangle Isosceles triangle theorem Isotomic conjugate Isotomic lines Jacobi point Japanese
List_of_triangle_topics
Concept in mathematical analysis
restating the problem as a minimization of the Rayleigh quotient. The isoperimetric inequality can be deduced from the Pólya–Szegő inequality with p = 1
Pólya–Szegő_inequality
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states
Gaussian isoperimetric inequality
Gaussian_isoperimetric_inequality
Concept in topology
In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold
Isoperimetric_dimension
Non-Euclidean geometry
(n-1)} -sphere of radius 1. The hyperbolic space also satisfies a linear isoperimetric inequality, that is there exists a constant i {\displaystyle i} such
Hyperbolic_space
Concept in geometry
least perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean
Area_of_a_circle
Concept in mathematics
linear isoperimetric inequality; it turns out that having such an isoperimetric inequality characterises Gromov-hyperbolic spaces. Linear isoperimetric inequalities
Hyperbolic_metric_space
Number, approximately 3.14
William (1894). "Isoperimetrical problems". Nature Series: Popular Lectures and Addresses. II: 571–592. Chavel, Isaac (2001). Isoperimetric inequalities.
Pi
Path that surrounds an area
the figures have the same convex hull; the big, first hexagon. The isoperimetric problem is to determine a figure with the largest area, amongst those
Perimeter
Notion in statistics
The Fisher information matrix plays a role in an inequality like the isoperimetric inequality. Of all probability distributions with a given entropy, the
Fisher_information
Theorem in geometry
1 ) {\textstyle c(X)={\frac {\mu (K)^{1/n}}{S(K)^{1/(n-1)}}}} . The isoperimetric inequality states that this is maximized on Euclidean balls. The Brunn–Minkowski
Brunn–Minkowski_theorem
Topics referred to by the same term
Dido, Texas, a ghost town in Tarrant County, Texas Dido's problem, the isoperimetric problem in mathematics All pages with titles containing dido This disambiguation
Dido_(disambiguation)
Study of geometric properties of sets through measure theory
generalizes and adapts Fubini's theorem to geometric measure theory. The isoperimetric inequality, which states that the smallest possible circumference for
Geometric_measure_theory
Ancient Greek mathematician (c. 200–140 BC)
rays reflected from it meet a point and thus cause burning. Zenodorus is known for authoring the treatise On isoperimetric figures, now lost. Many of its
Zenodorus_(mathematician)
Form of political manipulation
subdivisions, such as neighborhoods or voting districts (something isoperimetric rules would discourage); and it allows concave coastline districts,
Gerrymandering
Simple curve of Euclidean geometry
relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality. If a circle of radius r is centred at the vertex of an angle
Circle
Swiss mathematician (1796–1863)
the famous Steiner's chain of tangential circles, and a proof of the isoperimetric theorem (later a flaw was found in the proof, but was corrected by Weierstrass)
Jakob_Steiner
American mathematician
Sobolev inequalities, Hoffman and Spruck were also able to derive new isoperimetric inequalities for submanifolds of Riemannian manifolds.[HS74] It is well
David_Allen_Hoffman
Sequences of convex sets in a bounded set have convergent subsequences
the unit ball has a limit point (and that limit point is itself a compact set). As an example of its use, the isoperimetric problem can be shown to have
Blaschke_selection_theorem
Triangle with at least two sides congruent
{\displaystyle T} and perimeter p {\displaystyle p} are related by the isoperimetric inequality p 2 > 12 3 T . {\displaystyle p^{2}>12{\sqrt {3}}T.} This
Isosceles_triangle
Set of points equidistant from a center
the sphere is the one having the greatest volume. It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen
Sphere
Distance function defined between probability distributions
theory to optimal transport with quadratic cost. At this point, a very short proof of the isoperimetric inequality appears. The inequality states that among
Wasserstein_metric
Statistical parameter
The first example goes back to Paul Lévy. According to the spherical isoperimetric inequality, among all subsets A {\displaystyle A} of the sphere S n
Concentration_of_measure
Inequality in differential geometry
1983). Pu's inequality bears a curious resemblance to the classical isoperimetric inequality L 2 ≥ 4 π A {\displaystyle L^{2}\geq 4\pi A} for Jordan curves
Pu's_inequality
T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles: T ≤ 3 36 ( a + b + c ) 2 = 3 9 s 2 {\displaystyle
List_of_triangle_inequalities
two umbilical points. Cartan–Hadamard conjecture: can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Property of a mathematical space
Hurst exponent Isoperimetric dimension Metric dimension Order dimension q-dimension Fractal (q = 1) Correlation (q = 2) 0 dimension Point Zero-dimensional
Dimension
Linear algebra aspects of graph theory
eigenvalue of its Laplacian. The Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph
Spectral_graph_theory
Formula to quantify column buckling under a given load
lines enjoying the maximum-minimum property, or the solution of the isoperimetric problem in the broadest sense] (in Latin). Geneva, Switzerland: Marc
Euler's_critical_load
Gaussian copula Gaussian measure Gaussian correlation inequality Gaussian isoperimetric inequality Gauss's inequality Gauss-Helmert model The normal distribution
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Result in geometry
| {\displaystyle 2|\pi _{j}(E)|\leq |\partial E|} , we get a loose isoperimetric inequality: | E | d − 1 ≤ 2 − d | ∂ E | d {\displaystyle |E|^{d-1}\leq
Loomis–Whitney_inequality
Swiss mathematician (1707–1783)
curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense) Introductio in analysin infinitorum
Leonhard_Euler
gravitation, Hele-Shaw flows of viscous fluids, and purely mathematical isoperimetric problems, and interest in them seems to be steadily growing. They were
Quadrature_domains
Topics referred to by the same term
dimension Inductive dimension Lebesgue covering dimension Packing dimension Isoperimetric dimension Measurements of objects can be referred to as dimensions Dimensions
Dimension_(disambiguation)
Overview of and topical guide to geometry
system Monge's theorem Power center Nine-point circle Circle points segments proof Mrs. Miniver's problem Isoperimetric theorem Annulus Ptolemaios' theorem
Outline_of_geometry
French mathematician (born 1952)
methods to bound stochastic processes. He discovered new aspects of the isoperimetric and concentration of measure phenomena for product spaces, by obtaining
Michel_Talagrand
Baltic German mathematician
recompensed but I am still grateful to Hitler". Chebyshev function Isoperimetric inequality Low-rank approximation List of Baltic German scientists Erhard
Erhard_Schmidt
Second-smallest eigenvalue of a graph Laplacian
connectivity also relates to other connectivity attributes, such as the isoperimetric number, which is bounded below by half the algebraic connectivity. The
Algebraic_connectivity
Italian-French scientist (1736–1813)
ISBN 978-1-4357-1633-9. Although some authors speak of a general method of solving "isoperimetric problems", the eighteenth-century meaning of this expression amounts
Joseph-Louis_Lagrange
Size of a two-dimensional surface
be calculated using the "Surveyor's formula" (shoelace formula). The isoperimetric inequality states that, for a closed curve of length L (so the region
Area
Plane figure bounded by line segments
number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds. For any two
Polygon
Quadrilateral with four right angles
when ℓ = w {\displaystyle \ell =w\,} , the rectangle is a square. The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter
Rectangle
Type of group used in topology and geometric group theory
properly by isometries on a CAT(0) space, although they have quadratic isoperimetric inequality. Automorphism groups of free groups of rank ≥ 3 {\displaystyle
CAT(0)_group
Israeli mathematician of Latvian origin (1948–2024)
JSTOR 2951832. Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu (2002). "Isoperimetric and isodiametric functions of groups". Annals of Mathematics. 2. 156
Eliyahu_Rips
Shape with three equal sides
be obtained by substituting the altitude formula. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among
Equilateral_triangle
Triangle with circular arc edges
Möbius transformations. Circular triangles give the solution to an isoperimetric problem in which one seeks a curve of minimum length that encloses three
Circular_triangle
(operator theory) Caristi fixed-point theorem (fixed points) Envelope theorem (calculus of variations) Isoperimetric theorem (curves, calculus of variations)
List_of_theorems
Motion of a curve based on its curvature
decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape
Curve-shortening_flow
Inequality in Riemannian geometry
mapping a point p of X, to the real function on X given by the distance from the point p. The proof utilizes the coarea inequality, the isoperimetric inequality
Gromov's systolic inequality for essential manifolds
Gromov's_systolic_inequality_for_essential_manifolds
Largest distance between two points
isodiametric inequality or Bieberbach inequality, a relative of the isoperimetric inequality, states that, for a given diameter, the planar shape with
Diameter_of_a_set
American mathematician
generalized submanifolds. Moreover, they identified new results on the isoperimetric problem and its relation to the Sobolev embedding theorem. Their paper
Herbert_Federer
Process forming a path from many random steps
network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev
Random_walk
Chinese-American mathematician (born 1949)
assumptions. Around the same time, a similar inequality was obtained by isoperimetric methods by Mikhael Gromov, although his result is weaker than Li and
Shing-Tung_Yau
geometric measure theory concerning the generalization of the classical isoperimetric inequality to spaces of nonpositive sectional curvature Cauchy–Hadamard
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Four-sided polygon
perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality
Quadrilateral
Branch of mathematics
Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the
Geometry
cryptography Inflection point Inscribed square problem intercept, y-intercept, x-intercept Intersection number Intrinsic equation Isoperimetric inequality Jordan
List_of_curves_topics
Shape with same width in all directions
{\displaystyle \pi d} for the perimeter of a circle given its diameter. By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of
Curve_of_constant_width
Cubic plane curve
2008, pp. 402–403, Lemma 18.1. Monterde, J.; Ongay, F. (2012), "An isoperimetric type problem for primitive Pythagorean hodograph curves", Computer Aided
Tschirnhausen_cubic
Mathematics of smooth surfaces
Fernando Codá Marques and André Neves. Isoperimetric inequalities. In 1939 Schmidt proved that the classical isoperimetric inequality for curves in the Euclidean
Differential geometry of surfaces
Differential_geometry_of_surfaces
Homotopic map of a graph
automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem
Train_track_map
Partitioning a digital image into segments
algorithms of this category are normalized cuts, random walker, minimum cut, isoperimetric partitioning, minimum spanning tree-based segmentation, and segmentation-based
Image_segmentation
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
arbelos. Book V discusses isoperimetric figures, summarizing otherwise lost works by Zenodotus and Archimedes on isoperimetric plane and solid figures,
Ancient_Greek_mathematics
Uniform tiling of the plane using regular polygons
Tessellations of regular polygons that contain more than one type of vertex point Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings"
33344-33434_tiling
Perimeter of a circle or ellipse
two-dimensional surface Circumgon – Geometric figure which circumscribes a circle Isoperimetric inequality – Geometric inequality applicable to any closed curve Perimeter-equivalent
Circumference
Riemannian metric. Isometry is a surjective map which preserves distances. Isoperimetric function of a metric space X {\textstyle X} measures "how efficiently
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Parabolic partial differential equation
are the critical points for the mean curvature flow; minima solve the isoperimetric problem. For manifolds embedded in a Kähler–Einstein manifold, if the
Mean_curvature_flow
British mathematician
theorems for Lipschitz functions not reducible to one-point extensions and solved the reverse isoperimetric problem. He produced a sharp version of the Banach-Steinhaus
Keith_Martin_Ball
Shape with four equal sides and angles
area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds: 16 A ≤ P 2 {\displaystyle 16A\leq P^{2}} with equality
Square
Type of mathematical plane curve
caustic. The oriented area of the Wigner caustic improves the classical isoperimetric inequality. A non-singular hedgehog has a unique tangent line in each
Hedgehog_(geometry)
Form of differential geometry
of as analogous to Bonnesen's inequality with isoperimetric defect, a strengthening of the isoperimetric inequality. A number of new inequalities of this
Systolic_geometry
Existence of geodesic circles on surfaces
David; Gnepp, Andrei; Ng, Ting; Spivack, John; Yoder, Cara (2005), "The isoperimetric problem on some singular surfaces", Journal of the Australian Mathematical
Theorem of the three geodesics
Theorem_of_the_three_geodesics
can then be thought of as the isosystolic defect, analogous to the isoperimetric defect of Bonnesen's inequality. This approach therefore produces the
Loewner's_torus_inequality
Russian-French mathematician
nonvanishing sequence of sets can be metrically thickened to include almost every point. This closely mimics the phenomena of the law of large numbers, and in fact
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
expansion is bounded away from zero. expansion 1. The edge expansion, isoperimetric number, or Cheeger constant of a graph G is the minimum ratio, over
Glossary_of_graph_theory
It is a standard tool in integral geometry and has applications in isoperimetric and rigidity results. The formula is named after Luis Santaló, who first
Santaló's_formula
Inequality applying to triangles
S={\frac {a^{2}+b^{2}+c^{2}}{4\Delta }}} . List of triangle inequalities Isoperimetric inequality Hadwiger–Finsler inequality Claudi Alsina, Roger B. Nelsen:
Weitzenböck's_inequality
Differential calculus on function spaces
brachistochrone problem Solution to the tautochrone problem Solution to isoperimetric problems Calculating geodesics Finding minimal surfaces and solving
Calculus_of_variations
Differentialgeometrie I. Berlin: Springer-Verlag. Kazdan, Jerry L. (1982). "An isoperimetric inequality and Wiedersehen manifolds". Seminar on Differential Geometry
Wiedersehen_pair
Israeli mathematician and professor
astonishing link between Mahler's conjecture in convexity theory and an isoperimetric-type inequality involving symplectic capacities (with R. Karasev and
Shiri_Artstein
Description in Riemannian geometry
manifolds. The Cartan–Hadamard conjecture states that the classical isoperimetric inequality should hold in all simply connected spaces of non-positive
Sectional_curvature
mentioned above, is diffusion. Among others are: the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central
Uses_of_trigonometry
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
MR 2231924 Andrews, Ben; Bryan, Paul (2010), "Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere", Calc. Var.
Uniformization_theorem
Swedish Mathematician
in particular the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities (1960), his counterexamples related to Hilbert's
Vladimir_Mazya
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
Study of Boolean functions via discrete Fourier analysis
of this theorem used the invariance principle in conjunction with an isoperimetric theorem of Borell in Gaussian space; since then more direct proofs were
Analysis_of_Boolean_functions
Mathematical concept
consequence of the Tits alternative). Hyperbolic groups satisfy a linear isoperimetric inequality. Hyperbolic groups are always finitely presented. In fact
Hyperbolic_group
Greek mathematician (1873–1950)
Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems. Previously, between the mid-1700s to the mid-1800s, Leonhard
Constantin_Carathéodory
Gauss–Markov process Gaussian process Gaussian random field Gaussian isoperimetric inequality Large deviations of Gaussian random functions Girsanov's
List_of_probability_topics
Catherine Bandle (born 1943), Swiss expert on differential equations and isoperimetric inequalities Selenne Bañuelos (born 1985), Mexican-American mathematician
List_of_women_in_mathematics
On least area of curves of constant width
shots by O ( log log n ) {\displaystyle O(\log \log n)} . By the isoperimetric inequality, the curve of constant width in the Euclidean plane with
Blaschke–Lebesgue_theorem
Differentiable manifold
Mathematical Society. pp. 143–152. Chanillo, Sagun; Yang, Paul C. (2009). "Isoperimetric and Volume Comparison theorems on CR manifolds". Annali della Scuola
CR_manifold
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