Isoperimetric inequality

Isoperimetric inequality (Redirected from Isoperimetric theorem) Ir para a navegação Ir para a pesquisa Em matemática, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. Dentro {estilo de exibição m} -dimensional space {estilo de exibição mathbb {R} ^{n}} the inequality lower bounds the surface area or perimeter {nome do operador de estilo de exibição {por} (S)} of a set {displaystyle Ssubset mathbb {R} ^{n}} by its volume {nome do operador de estilo de exibição {volume} (S)} , {nome do operador de estilo de exibição {por} (S)geq noperatorname {volume} (S)^{fratura {n-1}{n}},nome do operador {volume} (B_{1})^{fratura {1}{n}}} , Onde {estilo de exibição B_{1}subconjunto mathbb {R} ^{n}} is a unit sphere. The equality holds only when {estilo de exibição S} is a sphere in {estilo de exibição mathbb {R} ^{n}} .

On a plane, ou seja. quando {estilo de exibição n=2} , 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 {estilo de exibição mathbb {R} ^{2}} , the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, este {estilo de exibição L^{2}geq 4pi A,} 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.[1] 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. No entanto, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Desde então, many other proofs have been found.

The isoperimetric problem has been extended in multiple ways, por exemplo, 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. Nomeadamente, 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.

Conteúdo 1 The isoperimetric problem in the plane 2 On a plane 3 On a sphere 4 In Rn 5 In Hadamard manifolds 6 In a metric measure space 7 For graphs 7.1 Exemplo: Isoperimetric inequalities for hypercubes 7.1.1 Edge isoperimetric inequality 7.1.2 Vertex isoperimetric inequality 8 Isoperimetric inequality for triangles 9 Veja também 10 Notas 11 Referências 12 External links The isoperimetric problem in the plane If a region is not convex, uma "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged. An elongated shape can be made more round while keeping its perimeter fixed and increasing its area.

The classical isoperimetric problem dates back to antiquity.[2] The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (caso existam) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (caso existam) minimizes the perimeter?

This problem is conceptually related to the principle of least action in physics, in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal Nicholas of Cusa, considered rotational action, the process by which a circle is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum (The Sacred Mystery of the Cosmos, 1596).

Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation.[3] Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

Steiner begins with some geometric constructions which are easily understood; por exemplo, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, por "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links).

On a plane The solution to the isoperimetric problem is usually expressed in the form of an inequality that relates the length L of a closed curve and the area A of the planar region that it encloses. The isoperimetric inequality states that {displaystyle 4pi Aleq L^{2},} and that the equality holds if and only if the curve is a circle. The area of a disk of radius R is πR2 and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π2R2 in this case.

Dozens of proofs of the isoperimetric inequality have been found. Dentro 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves (not assumed to be smooth). An elegant direct proof based on comparison of a smooth simple closed curve with an appropriate circle was given by E. Schmidt in 1938. It uses only the arc length formula, expression for the area of a plane region from Green's theorem, and the Cauchy–Schwarz inequality.

For a given closed curve, the isoperimetric quotient is defined as the ratio of its area and that of the circle having the same perimeter. This is equal to {displaystyle Q={fratura {4pi A}{L^{2}}}} and the isoperimetric inequality says that Q ≤ 1. Equivalentemente, the isoperimetric ratio L2/A is at least 4π for every curve.

The isoperimetric quotient of a regular n-gon is {displaystyle Q_{n}={fratura {pi }{ntan {tfrac {pi }{n}}}}.} Deixar {estilo de exibição C} be a smooth regular convex closed curve. Then the improved isoperimetric inequality states the following {estilo de exibição L^{2}geqslant 4pi A+8pi left|{widetilde {UMA}}_{0.5}certo|,} Onde {estilo de exibição L,UMA,{widetilde {UMA}}_{0.5}} denote the length of {estilo de exibição C} , the area of the region bounded by {estilo de exibição C} and the oriented area of the Wigner caustic of {estilo de exibição C} , respectivamente, and the equality holds if and only if {estilo de exibição C} is a curve of constant width.[4] On a sphere Let C be a simple closed curve on a sphere of radius 1. Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that {estilo de exibição L^{2}geq A(4pi -A),} and that the equality holds if and only if the curve is a circle. Há, na verdade, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.

This inequality was discovered by Paul Lévy (1919) who also extended it to higher dimensions and general surfaces.[5] In the more general case of arbitrary radius R, it is known [6] este {estilo de exibição L^{2}geq 4pi A-{fratura {A^{2}}{R^{2}}}.} In Rn The isoperimetric inequality states that a sphere has the smallest surface area per given volume. Given a bounded set {displaystyle Ssubset mathbb {R} ^{n}} with surface area {nome do operador de estilo de exibição {por} (S)} and volume {nome do operador de estilo de exibição {volume} (S)} , the isoperimetric inequality states {nome do operador de estilo de exibição {por} (S)geq noperatorname {volume} (S)^{fratura {n-1}{n}},nome do operador {volume} (B_{1})^{fratura {1}{n}},} Onde {estilo de exibição B_{1}subconjunto mathbb {R} ^{n}} is a unit ball. The equality holds when {estilo de exibição S} is a ball in {estilo de exibição mathbb {R} ^{n}} . Under additional restrictions on the set (such as convexity, regularity, smooth boundary), the equality holds for a ball only. But in full generality the situation is more complicated. The relevant result of Schmidt (1949, Sect. 20.7) (for a simpler proof see Baebler (1957)) is clarified in Hadwiger (1957, Sect. 5.2.5) do seguinte modo. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. Aquilo é, the equality holds for a compact set {estilo de exibição S} se e apenas se {estilo de exibição S} contains a closed ball {estilo de exibição B} de tal modo que {nome do operador de estilo de exibição {volume} (B)=nome do operador {volume} (S)} e {nome do operador de estilo de exibição {por} (B)=nome do operador {por} (S).} Por exemplo, a "corona" may be a curve.

The proof of the inequality follows directly from Brunn–Minkowski inequality between a set {estilo de exibição S} and a ball with radius {displaystyle épsilon } , ou seja. {estilo de exibição B_{épsilon }=epsilon B_{1}} . By taking Brunn–Minkowski inequality to the power {estilo de exibição m} , subtracting {nome do operador de estilo de exibição {volume} (S)} from both sides, dividing them by {displaystyle épsilon } , and taking the limit as {displaystyle epsilon to 0.} (Osserman (1978); Federer (1969, §3.2.43)).

In full generality (Federer 1969, §3.2.43), the isoperimetric inequality states that for any set {displaystyle Ssubset mathbb {R} ^{n}} whose closure has finite Lebesgue measure {estilo de exibição m,ômega _{n}^{fratura {1}{n}}L^{n}({bar {S}})^{fratura {n-1}{n}}leq M_{*}^{n-1}(partial S)} Onde {estilo de exibição M_{*}^{n-1}} é o (n-1)-dimensional Minkowski content, Ln is the n-dimensional Lebesgue measure, and ωn is the volume of the unit ball in {estilo de exibição mathbb {R} ^{n}} . If the boundary of S is rectifiable, then the Minkowski content is the (n-1)-dimensional Hausdorff measure.

The n-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the Sobolev inequality on {estilo de exibição mathbb {R} ^{n}} with optimal constant: {estilo de exibição à esquerda(int_{mathbb {R} ^{n}}|você|^{fratura {n}{n-1}}certo)^{fratura {n-1}{n}}leq n^{-1}ômega _{n}^{-{fratura {1}{n}}}int_{mathbb {R} ^{n}}|nabla u|} para todos {estilo de exibição uin W ^{1,1}(mathbb {R} ^{n})} .

In Hadamard manifolds Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space {estilo de exibição mathbb {R} ^{n}} , which is a Hadamard manifold with curvature zero. In 1970's and early 80's, Thierry Aubin, Misha Gromov, Yuri Burago, and Viktor Zalgaller conjectured that the Euclidean isoperimetric inequality {nome do operador de estilo de exibição {por} (S)geq noperatorname {volume} (S)^{fratura {n-1}{n}}nome do operador {volume} (B_{1})^{fratura {1}{n}}} holds for bounded sets {estilo de exibição S} in Hadamard manifolds, which has become known as the Cartan–Hadamard conjecture. In dimension 2 this had already been established in 1926 by André Weil, who was a student of Hadamard at the time. In dimensions 3 e 4 the conjecture was proved by Bruce Kleiner in 1992, and Chris Croke in 1984 respectivamente.

In a metric measure space Most of the work on isoperimetric problem has been done in the context of smooth regions in Euclidean spaces, or more generally, in Riemannian manifolds. No entanto, the isoperimetric problem can be formulated in much greater generality, using the notion of Minkowski content. Deixar {estilo de exibição (X,dentro ,d)} be a metric measure space: X is a metric space with metric d, and μ is a Borel measure on X. The boundary measure, or Minkowski content, of a measurable subset A of X is defined as the lim inf {displaystyle mu ^{+}(UMA)=liminf_{varepsilon para 0+}{fratura {dentro (UMA_{varepsilon })-dentro (UMA)}{varepsilon }},} Onde {estilo de exibição A_{varepsilon }={xin X|d(x,UMA)leq varepsilon }} is the ε-extension of A.

The isoperimetric problem in X asks how small can {displaystyle mu ^{+}(UMA)} be for a given μ(UMA). If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.

A função {estilo de exibição I(uma)=inf{mu ^{+}(UMA)|dentro (UMA)=a}} is called the isoperimetric profile of the metric measure space {estilo de exibição (X,dentro ,d)} . Isoperimetric profiles have been studied for Cayley graphs of discrete groups and for special classes of Riemannian manifolds (where usually only regions A with regular boundary are considered).

For graphs Main article: Expander graph In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.[7] Isoperimetric inequalities for graphs relate the size of vertex subsets to the size of their boundary, which is usually measured by the number of edges leaving the subset (edge expansion) or by the number of neighbouring vertices (vertex expansion). For a graph {estilo de exibição G} and a number {estilo de exibição k} , the following are two standard isoperimetric parameters for graphs.[8] The edge isoperimetric parameter: {displaystyle Phi _{E}(G,k)=min _{Ssubseteq V}deixei{|E(S,{overline {S}})|:|S|=kright}} The vertex isoperimetric parameter: {displaystyle Phi _{V}(G,k)=min _{Ssubseteq V}deixei{|Gama (S)setminus S|:|S|=kright}} Aqui {estilo de exibição E(S,{overline {S}})} denotes the set of edges leaving {estilo de exibição S} e {displaystyle Gama (S)} denotes the set of vertices that have a neighbour in {estilo de exibição S} . The isoperimetric problem consists of understanding how the parameters {displaystyle Phi _{E}} e {displaystyle Phi _{V}} behave for natural families of graphs.

Exemplo: Isoperimetric inequalities for hypercubes The {estilo de exibição d} -dimensional hypercube {displaystyle Q_{d}} is the graph whose vertices are all Boolean vectors of length {estilo de exibição d} , isso é, the set {estilo de exibição {0,1}^{d}} . Two such vectors are connected by an edge in {displaystyle Q_{d}} if they are equal up to a single bit flip, isso é, their Hamming distance is exactly one. The following are the isoperimetric inequalities for the Boolean hypercube.[9] Edge isoperimetric inequality The edge isoperimetric inequality of the hypercube is {displaystyle Phi _{E}(Q_{d},k)geq k(d-log _{2}k)} . This bound is tight, as is witnessed by each set {estilo de exibição S} that is the set of vertices of any subcube of {displaystyle Q_{d}} .

Vertex isoperimetric inequality Harper's theorem[10] says that Hamming balls have the smallest vertex boundary among all sets of a given size. Hamming balls are sets that contain all points of Hamming weight at most {estilo de exibição r} and no points of Hamming weight larger than {displaystyle r+1} for some integer {estilo de exibição r} . This theorem implies that any set {displaystyle Ssubseteq V} com {estilo de exibição |S|soma geq _{i=0}^{r}{d choose i}} satisfies {estilo de exibição |Scup Gamma (S)|soma geq _{i=0}^{r+1}{d choose i}.} [11] As a special case, consider set sizes {displaystyle k=|S|} of the form {displaystyle k={d choose 0}+{d choose 1}+pontos +{d choose r}} for some integer {estilo de exibição r} . Then the above implies that the exact vertex isoperimetric parameter is {displaystyle Phi _{V}(Q_{d},k)={d choose r+1}.} [12] Isoperimetric inequality for triangles The isoperimetric inequality for triangles in terms of perimeter p and area T states that[13][14] {estilo de exibição p^{2}geq 12{quadrado {3}}cdot T,} with equality for the equilateral triangle. This is implied, via the AM–GM inequality, by a stronger inequality which has also been called the isoperimetric inequality for triangles:[15] {displaystyle Tleq {fratura {quadrado {3}}{4}}(abc)^{fratura {2}{3}}.} See also Mathematics portal Blaschke–Lebesgue theorem Chaplygin problem Curve-shortening flow Expander graph Gaussian isoperimetric inequality Isoperimetric dimension Isoperimetric point List of triangle inequalities Planar separator theorem Mixed volume Notes ^ Blåsjö, Viktor (2005). "The Evolution of the Isoperimetric Problem". América. Matemática. Por mês. 112 (6): 526–566. doi:10.2307/30037526. JSTOR 30037526. ^ Olmo, Carlos Beltrán, Irene (4 Janeiro 2021). "Sobre mates y mitos". EL PAÍS (in Spanish). Recuperado 14 Janeiro 2021. ^ J. Steiner, Einfacher Beweis der isoperimetrischen Hauptsätze, J. reine angew Math. 18, (1838), pp. 281–296; and Gesammelte Werke Vol. 2, pp. 77–91, Reimer, Berlim, (1882). ^ Zwierzyński, Michał (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals". J. Matemática. Anal. Appl. 442 (2): 726–739. arXiv:1512.06684. doi:10.1016/j.jmaa.2016.05.016. S2CID 119708226. ^ Gromov, Mikhail; Pansu, Pierre (2006). "Appendix C. Paul Levy's Isoperimetric Inequality". Metric Structures for Riemannian and Non-Riemannian Spaces. Clássicos modernos de Birkhäuser. Dordrecht: Springer. p. 519. ISBN 9780817645830. ^ Osserman, Roberto. "The Isoperimetric Inequality." Boletim da American Mathematical Society. 84.6 (1978) http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf ^ Hoory, Linial & Widgerson (2006) ^ Definitions 4.2 e 4.3 of Hoory, Linial & Widgerson (2006) ^ See Bollobás (1986) and Section 4 in Hoory, Linial & Widgerson (2006) ^ Cf. Calabro (2004) or Bollobás (1986) ^ cf. Leader (1991) ^ Also stated in Hoory, Linial & Widgerson (2006) ^ Chakerian, G. D. "A Distorted View of Geometry." CH. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Associação Matemática da América, 1979: 147. ^ "The isoperimetric inequality for triangles". ^ Dragutin Svrtan and Darko Veljan, "Non-Euclidean Versions of Some Classical Triangle Inequalities", Fórum geométrico 12, 2012, 197–209. http://forumgeom.fau.edu/FG2012volume12/FG201217.pdf References Blaschke and Leichtweiß, Elementare Differentialgeometrie (em alemão), 5th edition, completely revised by K. Leichtweiß. Os ensinamentos básicos das ciências matemáticas, Band 1. Springer-Verlag, New York Heidelberg Berlin, 1973 ISBN 0-387-05889-3 Bollobás, Béla (1986). Combinatória: set systems, hypergraphs, families of vectors, and combinatorial probability. Cambridge University Press. ISBN 978-0-521-33703-8. Burago (2001) [1994], "Isoperimetric inequality", Enciclopédia de Matemática, EMS Press Calabro, Chris (2004). "Harper's Theorem" (PDF). Recuperado 8 Fevereiro 2011. Capogna, Luca; Donatella Danielli; Scott Pauls; Jeremy Tyson (2007). An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Birkhäuser Verlag. ISBN 978-3-7643-8132-5. Fenchel, Werner; Bonnesen, Tommy (1934). Theorie der konvexen Körper. Resultados de matemática e suas áreas de fronteira. Volume. 3. Berlim: 1. Verlag von Julius Springer. Fenchel, Werner; Bonnesen, Tommy (1987). Theory of convex bodies. Moscou, Idaho: eu. Boron, C. Christenson and B. Smith. BCS Associates. Federer, Herbert (1969). Geometric measure theory. Springer-Verlag. ISBN 3-540-60656-4.. Gromov, M.: "Paul Levy's isoperimetric inequality". Appendix C in Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, Massachusetts, 1999. Hadwiger, Hugo (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag.. Hoory, Shlomo; linear, Natan; Widgerson, Avi (2006). "Expander graphs and their applications" (PDF). Boletim da American Mathematical Society. Nova série. 43 (4): 439–561. doi:10.1090/S0273-0979-06-01126-8. Leader, Imre (1991). "Discrete isoperimetric inequalities". Proceedings of Symposia in Applied Mathematics. Volume. 44. pp. 57–80. Osserman, Roberto (1978). "The isoperimetric inequality". Touro. América. Matemática. Soc. 84 (6): 1182–1238. doi:10.1090/S0002-9904-1978-14553-4.. Zwierzyński, Michał (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals". J. Matemática. Anal. Appl. 442 (2): 726–739. arXiv:1512.06684. doi:10.1016/j.jmaa.2016.05.016. S2CID 119708226. Schmidt, Erhard (1949). "Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Hugel in der euklidischen und nichteuklidischen Geometrie. II". Matemática. Nachr. 2 (3-4): 171-244. doi:10.1002/mana.19490020308.. Baebler, F. (1957). "Zum isoperimetrischen Problem". Arch. Matemática. (Basileia). 8: 52–65. doi:10.1007/BF01898439. S2CID 123704157.. External links Wikimedia Commons has media related to Isoperimetric inequality. History of the Isoperimetric Problem at Convergence Treiberg: Several proofs of the isoperimetric inequality Isoperimetric Theorem at cut-the-knot Categories: Multivariable calculusCalculus of variationsGeometric inequalitiesAnalytic geometry