Misplaced Pages

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77-454: 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

154-407: A b | f ′ ( t ) |   d t . {\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}=\lim _{N\to \infty }\sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt.} The last equality is proved by the following steps: With

231-899: A b | g ′ ( φ ( t ) ) φ ′ ( t ) |   d t = ∫ a b | g ′ ( φ ( t ) ) | φ ′ ( t )   d t in the case  φ  is non-decreasing = ∫ c d | g ′ ( u ) |   d u using integration by substitution = L ( g ) . {\displaystyle {\begin{aligned}L(f)&=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt=\int _{a}^{b}{\Big |}g'(\varphi (t))\varphi '(t){\Big |}\ dt\\&=\int _{a}^{b}{\Big |}g'(\varphi (t)){\Big |}\varphi '(t)\ dt\quad {\text{in

308-564: A ) {\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)} , and N > ( b − a ) / δ ( ε ) {\displaystyle N>(b-a)/\delta (\varepsilon )} . In the limit N → ∞ , {\displaystyle N\to \infty ,} δ ( ε ) → 0 {\displaystyle \delta (\varepsilon )\to 0} so ε → 0 {\displaystyle \varepsilon \to 0} thus

385-1301: A ) / δ ( ε ) {\textstyle N>(b-a)/\delta (\varepsilon )} so that Δ t < δ ( ε ) {\displaystyle \Delta t<\delta (\varepsilon )} , it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − | f ′ ( t i ) | ) < ε N Δ t {\displaystyle \Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\left|f'(t_{i})\right|\right)<\varepsilon N\Delta t} with | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\displaystyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } , ε N Δ t = ε ( b −

462-509: A + i ( b − a ) / N = a + i Δ t {\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t} with Δ t = b − a N = t i − t i − 1 {\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}} for i = 0 , 1 , … , N . {\displaystyle i=0,1,\dotsc ,N.} This definition

539-661: A , b ] → R n {\displaystyle f:[a,b]\to \mathbb {R} ^{n}} on [ a , b ] {\displaystyle [a,b]} is always finite, i.e., rectifiable . The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition L ( f ) = sup ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\sup \sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}} where

616-547: A finite number of points on the curve using (straight) line segments to create a polygonal path . Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summation of the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance . If

693-498: A sphere has the smallest surface area per given volume. Given a bounded open set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} with C 1 {\displaystyle C^{1}} boundary, having surface area per ⁡ ( S ) {\displaystyle \operatorname {per} (S)} and volume vol ⁡ ( S ) {\displaystyle \operatorname {vol} (S)} ,

770-416: A closed ball B {\displaystyle B} such that vol ⁡ ( B ) = vol ⁡ ( S ) {\displaystyle \operatorname {vol} (B)=\operatorname {vol} (S)} and per ⁡ ( B ) = per ⁡ ( S ) . {\displaystyle \operatorname {per} (B)=\operatorname {per} (S).} For example,

847-407: A closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals . In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of

SECTION 10

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924-817: A curve expressed in polar coordinates, the arc length is: ∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 d t = ∫ θ ( t 1 ) θ ( t 2 ) ( d r d θ ) 2 + r 2 d θ . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}\,}}dt=\int _{\theta (t_{1})}^{\theta (t_{2})}{\sqrt {\left({\frac {dr}{d\theta }}\right)^{2}+r^{2}\,}}d\theta .} The second expression

1001-403: A curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is x ( r , θ ) = ( r cos ⁡ θ , r sin ⁡ θ ) . {\displaystyle \mathbf {x} (r,\theta )=(r\cos \theta ,r\sin \theta ).} The integrand of the arc length integral

1078-788: A curve expressed in spherical coordinates where θ {\displaystyle \theta } is the polar angle measured from the positive z {\displaystyle z} -axis and ϕ {\displaystyle \phi } is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is x ( r , θ , ϕ ) = ( r sin ⁡ θ cos ⁡ ϕ , r sin ⁡ θ sin ⁡ ϕ , r cos ⁡ θ ) . {\displaystyle \mathbf {x} (r,\theta ,\phi )=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta ).} Using

1155-696: A curve expressed in spherical coordinates, the arc length is ∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 + r 2 sin 2 ⁡ θ ( d ϕ d t ) 2 d t . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+r^{2}\sin ^{2}\theta \left({\frac {d\phi }{dt}}\right)^{2}\,}}dt.} A very similar calculation shows that

1232-433: A finite limit as the lengths of the segments get arbitrarily small . For some curves, there is a smallest number L {\displaystyle L} that is an upper bound on the length of all polygonal approximations (rectification). These curves are called rectifiable and the arc length is defined as the number L {\displaystyle L} . A signed arc length can be defined to convey

1309-399: 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 and the isoperimetric inequality says that Q ≤ 1. Equivalently, the isoperimetric ratio L / A is at least 4 π for every curve. The isoperimetric quotient of a regular n -gon is Let C {\displaystyle C} be

1386-434: A quarter of the circle. Since d y / d x = − x / 1 − x 2 {\textstyle dy/dx=-x{\big /}{\sqrt {1-x^{2}}}} and 1 + ( d y / d x ) 2 = 1 / ( 1 − x 2 ) , {\displaystyle 1+(dy/dx)^{2}=1{\big /}\left(1-x^{2}\right),}

1463-481: A quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y = 1 − x 2 . {\displaystyle y={\sqrt {1-x^{2}}}.} The interval x ∈ [ − 2 / 2 , 2 / 2 ] {\displaystyle x\in \left[-{\sqrt {2}}/2,{\sqrt {2}}/2\right]} corresponds to

1540-441: A sense of orientation or "direction" with respect to a reference point taken as origin in the curve (see also: curve orientation and signed distance ). Let f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of

1617-509: 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

SECTION 20

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1694-415: A smooth regular convex closed curve. Then the improved isoperimetric inequality states the following where L , A , A ~ 0.5 {\displaystyle L,A,{\widetilde {A}}_{0.5}} denote the length of C {\displaystyle C} , the area of the region bounded by C {\displaystyle C} and the oriented area of

1771-443: 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; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which

1848-411: A special case, consider set sizes k = | S | {\displaystyle k=|S|} of the form for some integer r {\displaystyle r} . Then the above implies that the exact vertex isoperimetric parameter is The isoperimetric inequality for triangles in terms of perimeter p and area T states that with equality for the equilateral triangle . This

1925-711: Is ( x u u ′ + x v v ′ ) ⋅ ( x u u ′ + x v v ′ ) = g 11 ( u ′ ) 2 + 2 g 12 u ′ v ′ + g 22 ( v ′ ) 2 {\displaystyle \left(\mathbf {x} _{u}u'+\mathbf {x} _{v}v'\right)\cdot (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')=g_{11}\left(u'\right)^{2}+2g_{12}u'v'+g_{22}\left(v'\right)^{2}} (where g i j {\displaystyle g_{ij}}

2002-543: Is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} The chain rule for vector fields shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '.} So

2079-620: Is continuously differentiable , then it is simply a special case of a parametric equation where x = t {\displaystyle x=t} and y = f ( t ) . {\displaystyle y=f(t).} The Euclidean distance of each infinitesimal segment of the arc can be given by: d x 2 + d y 2 = 1 + ( d y d x ) 2 d x . {\displaystyle {\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.} The arc length

2156-430: Is another continuously differentiable parameterization of the curve originally defined by f . {\displaystyle f.} The arc length of the curve is the same regardless of the parameterization used to define the curve: L ( f ) = ∫ a b | f ′ ( t ) |   d t = ∫

2233-531: Is defined as the lim inf where is the ε- extension of A . The isoperimetric problem in X asks how small can μ + ( A ) {\displaystyle \mu ^{+}(A)} be for a given μ ( A ). 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

2310-597: Is equivalent to the standard definition of arc length as an integral: L ( f ) = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t = ∫

2387-529: Is exactly one. The following are the isoperimetric inequalities for the Boolean hypercube. The edge isoperimetric inequality of the hypercube is Φ E ( Q d , k ) ≥ k ( d − log 2 ⁡ k ) {\displaystyle \Phi _{E}(Q_{d},k)\geq k(d-\log _{2}k)} . This bound is tight, as is witnessed by each set S {\displaystyle S} that

Isoperimetric inequality - Misplaced Pages Continue

2464-420: Is for a polar graph r = r ( θ ) {\displaystyle r=r(\theta )} parameterized by t = θ {\displaystyle t=\theta } . Now let C ( t ) = ( r ( t ) , θ ( t ) , ϕ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))} be

2541-617: Is implied, via the AM–GM inequality , by a stronger inequality which has also been called the isoperimetric inequality for triangles: If and only if Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 251220261 Upstream caches: cp1108 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 11:03:25 GMT Arc length Arc length

2618-400: 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). 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

2695-505: Is not necessarily smooth, although the answer turns out to be the same. The function is called the isoperimetric profile of the metric measure space ( X , μ , d ) {\displaystyle (X,\mu ,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). In graph theory , isoperimetric inequalities are at

2772-426: Is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations. Let x ( u , v ) {\displaystyle \mathbf {x} (u,v)} be a surface mapping and let C ( t ) = ( u ( t ) , v ( t ) ) {\displaystyle \mathbf {C} (t)=(u(t),v(t))} be a curve on this surface. The integrand of

2849-669: Is the first fundamental form coefficient), so the integrand of the arc length integral can be written as g a b ( u a ) ′ ( u b ) ′ {\displaystyle {\sqrt {g_{ab}\left(u^{a}\right)'\left(u^{b}\right)'\,}}} (where u 1 = u {\displaystyle u^{1}=u} and u 2 = v {\displaystyle u^{2}=v} ). Let C ( t ) = ( r ( t ) , θ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t))} be

2926-441: Is the distance between two points along a section of a curve . Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification . For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e.,

3003-399: Is the graph whose vertices are all Boolean vectors of length d {\displaystyle d} , that is, the set { 0 , 1 } d {\displaystyle \{0,1\}^{d}} . Two such vectors are connected by an edge in Q d {\displaystyle Q_{d}} if they are equal up to a single bit flip, that is, their Hamming distance

3080-633: Is the set of vertices of any subcube of Q d {\displaystyle Q_{d}} . Harper's theorem 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 r {\displaystyle r} and no points of Hamming weight larger than r + 1 {\displaystyle r+1} for some integer r {\displaystyle r} . This theorem implies that any set S ⊆ V {\displaystyle S\subseteq V} with satisfies As

3157-890: Is the volume of the unit ball in R n {\displaystyle \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 R n {\displaystyle \mathbb {R} ^{n}} with optimal constant: for all u ∈ W 1 , 1 ( R n ) {\displaystyle u\in W^{1,1}(\mathbb {R} ^{n})} . Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize

Isoperimetric inequality - Misplaced Pages Continue

3234-455: Is then given by: s = ∫ a b 1 + ( d y d x ) 2 d x . {\displaystyle s=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.} Curves with closed-form solutions for arc length include the catenary , circle , cycloid , logarithmic spiral , parabola , semicubical parabola and straight line . The lack of

3311-552: 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 and 4 the conjecture was proved by Bruce Kleiner in 1992, and Chris Croke in 1984 respectively. 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 . However,

3388-481: The Wigner caustic of C {\displaystyle C} , respectively, and the equality holds if and only if C {\displaystyle C} is a curve of constant width . 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 and that the equality holds if and only if

3465-410: The supremum is taken over all possible partitions a = t 0 < t 1 < ⋯ < t N − 1 < t N = b {\displaystyle a=t_{0}<t_{1}<\dots <t_{N-1}<t_{N}=b} of [ a , b ] . {\displaystyle [a,b].} This definition as the supremum of

3542-431: The "corona" may be a curve. The proof of the inequality follows directly from Brunn–Minkowski inequality between a set S {\displaystyle S} and a ball with radius ϵ {\displaystyle \epsilon } , i.e. B ϵ = ϵ B 1 {\displaystyle B_{\epsilon }=\epsilon B_{1}} . By taking Brunn–Minkowski inequality to

3619-431: 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. The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in

3696-477: The Euclidean space R n {\displaystyle \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 holds for bounded sets S {\displaystyle S} in Hadamard manifolds, which has become known as

3773-2618: The above step result, it becomes ∑ i = 1 N | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | Δ t − ∑ i = 1 N | f ′ ( t i ) | Δ t . {\displaystyle \sum _{i=1}^{N}\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.} Terms are rearranged so that it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − ∫ 0 1 | f ′ ( t i ) | d θ ) ≦ Δ t ∑ i = 1 N ( ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) |   d θ − ∫ 0 1 | f ′ ( t i ) | d θ ) = Δ t ∑ i = 1 N ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) |   d θ {\displaystyle {\begin{aligned}&\Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad \leqq \Delta t\sum _{i=1}^{N}\left(\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|\ d\theta -\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad =\Delta t\sum _{i=1}^{N}\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\ d\theta \end{aligned}}} where in

3850-605: The all possible partition sums is also valid if f {\displaystyle f} is merely continuous, not differentiable. A curve can be parameterized in infinitely many ways. Let φ : [ a , b ] → [ c , d ] {\displaystyle \varphi :[a,b]\to [c,d]} be any continuously differentiable bijection . Then g = f ∘ φ − 1 : [ c , d ] → R n {\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}}

3927-792: The arc length integral is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} Evaluating the derivative requires the chain rule for vector fields: D ( x ∘ C ) = ( x u   x v ) ( u ′ v ′ ) = x u u ′ + x v v ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=(\mathbf {x} _{u}\ \mathbf {x} _{v}){\binom {u'}{v'}}=\mathbf {x} _{u}u'+\mathbf {x} _{v}v'.} The squared norm of this vector

SECTION 50

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4004-415: The area A of the planar region that it encloses. The isoperimetric inequality states that and that the equality holds if and only if the curve is a circle. The area of a disk of radius R is πR and the circumference of the circle is 2 πR , so both sides of the inequality are equal to 4 π R in this case. Dozens of proofs of the isoperimetric inequality have been found. In 1902, Hurwitz published

4081-432: The case }}\varphi {\text{ is non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\text{using integration by substitution}}\\&=L(g).\end{aligned}}} If a planar curve in R 2 {\displaystyle \mathbb {R} ^{2}} is defined by the equation y = f ( x ) , {\displaystyle y=f(x),} where f {\displaystyle f}

4158-686: The chain rule again shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ + x ϕ ϕ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '+\mathbf {x} _{\phi }\phi '.} All dot products x i ⋅ x j {\displaystyle \mathbf {x} _{i}\cdot \mathbf {x} _{j}} where i {\displaystyle i} and j {\displaystyle j} differ are zero, so

4235-685: The curve defined by f {\displaystyle f} can be defined as the limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity. This means L ( f ) = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}} where t i =

4312-422: The curve is a circle. There are, in fact, 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. In the more general case of arbitrary radius R , it is known that The isoperimetric inequality states that

4389-448: The curve is not already a polygonal path, then using a progressively larger number of line segments of smaller lengths will result in better curve length approximations. Such a curve length determination by approximating the curve as connected (straight) line segments is called rectification of a curve. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to

4466-434: The derivative is a continuous function) f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} , then the curve is rectifiable (i.e., it has a finite length). The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting

4543-578: 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) as follows. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. That is, the equality holds for a compact set S {\displaystyle S} if and only if S {\displaystyle S} contains

4620-427: 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

4697-399: 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 . Isoperimetric inequalities for graphs relate the size of vertex subsets to

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4774-442: The isoperimetric inequality states where B 1 ⊂ R n {\displaystyle B_{1}\subset \mathbb {R} ^{n}} is a unit ball . The equality holds when S {\displaystyle S} is a ball in R n {\displaystyle \mathbb {R} ^{n}} . Under additional restrictions on the set (such as convexity , regularity , smooth boundary ),

4851-419: The isoperimetric inequality states that for any set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} whose closure has finite Lebesgue measure where M ∗ n − 1 {\displaystyle M_{*}^{n-1}} is the ( n -1)-dimensional Minkowski content , L is the n -dimensional Lebesgue measure, and ω n

4928-401: The isoperimetric problem can be formulated in much greater generality, using the notion of Minkowski content . Let ( X , μ , d ) {\displaystyle (X,\mu ,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

5005-500: 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

5082-665: The left side of < {\displaystyle <} approaches 0 {\displaystyle 0} . In other words, ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t = ∑ i = 1 N | f ′ ( t i ) | Δ t {\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t} in this limit, and

5159-806: The leftmost side | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\textstyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } is used. By | | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) | | < ε {\textstyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon } for N > ( b −

5236-425: The length of a quarter of the unit circle is ∫ − 2 / 2 2 / 2 d x 1 − x 2 . {\displaystyle \int _{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}{\frac {dx}{\sqrt {1-x^{2}}}}\,.} The 15-point Gauss–Kronrod rule estimate for this integral of 1.570 796 326 808 177 differs from

5313-529: 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 . Steiner showed that if

5390-433: The plane of fixed perimeter, which curve (if any) 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 (if any) 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

5467-470: The power n {\displaystyle n} , subtracting vol ⁡ ( S ) {\displaystyle \operatorname {vol} (S)} from both sides, dividing them by ϵ {\displaystyle \epsilon } , and taking the limit as ϵ → 0. {\displaystyle \epsilon \to 0.} ( Osserman (1978) ; Federer (1969 , §3.2.43)). In full generality ( Federer 1969 , §3.2.43),

5544-452: The right side of this equality is just the Riemann integral of | f ′ ( t ) | {\displaystyle \left|f'(t)\right|} on [ a , b ] . {\displaystyle [a,b].} This definition of arc length shows that the length of a curve represented by a continuously differentiable function f : [

5621-647: The set of edges leaving S {\displaystyle S} and Γ ( S ) {\displaystyle \Gamma (S)} denotes the set of vertices that have a neighbour in S {\displaystyle S} . The isoperimetric problem consists of understanding how the parameters Φ E {\displaystyle \Phi _{E}} and Φ V {\displaystyle \Phi _{V}} behave for natural families of graphs. The d {\displaystyle d} -dimensional hypercube Q d {\displaystyle Q_{d}}

5698-486: 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 G {\displaystyle G} and a number k {\displaystyle k} , the following are two standard isoperimetric parameters for graphs. Here E ( S , S ¯ ) {\displaystyle E(S,{\overline {S}})} denotes

5775-937: The squared integrand of the arc length integral is ( x r ⋅ x r ) ( r ′ ) 2 + 2 ( x r ⋅ x θ ) r ′ θ ′ + ( x θ ⋅ x θ ) ( θ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 . {\displaystyle \left(\mathbf {x_{r}} \cdot \mathbf {x_{r}} \right)\left(r'\right)^{2}+2\left(\mathbf {x} _{r}\cdot \mathbf {x} _{\theta }\right)r'\theta '+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}.} So for

5852-1110: The squared norm of this vector is ( x r ⋅ x r ) ( r ′ 2 ) + ( x θ ⋅ x θ ) ( θ ′ ) 2 + ( x ϕ ⋅ x ϕ ) ( ϕ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 + r 2 sin 2 ⁡ θ ( ϕ ′ ) 2 . {\displaystyle \left(\mathbf {x} _{r}\cdot \mathbf {x} _{r}\right)\left(r'^{2}\right)+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}+\left(\mathbf {x} _{\phi }\cdot \mathbf {x} _{\phi }\right)\left(\phi '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}+r^{2}\sin ^{2}\theta \left(\phi '\right)^{2}.} So for

5929-446: The true length of arcsin ⁡ x | − 2 / 2 2 / 2 = π 2 {\displaystyle \arcsin x{\bigg |}_{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}={\frac {\pi }{2}}} by 1.3 × 10 and the 16-point Gaussian quadrature rule estimate of 1.570 796 326 794 727 differs from the true length by only 1.7 × 10 . This means it

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