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 resembles...

mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n {\displaystyle n} -dimensional space R...

spaces: Complex dimension Hausdorff dimension Inductive dimension Lebesgue covering dimension Packing dimension Isoperimetric dimension Measurements of...

Exterior dimension Hurst exponent Isoperimetric dimension Metric dimension Order dimension q-dimension Fractal (q = 1) Correlation (q = 2) 0 dimension Point...

shrinks to a point, the ratio becomes 4π. For higher-dimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as Bd/Vd − 1 where...

In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states...

Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the...

There are also higher-dimensional generalizations of isoperimetric and Dehn functions. For k ≥ 1 the k-dimensional isoperimetric function of a group bounds...

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal...

Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results...

network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev...

research focuses on analysis and geometry, including isoperimetric inequalities, curvature-dimension conditions, convex geometry, and concentration of measure...

S2CID 16836013. Croke, Christopher B. (1984). "A sharp four dimensional isoperimetric inequality". Commentarii Mathematici Helvetici. 59 (1): 187–192...

inequality is a functional isoperimetric inequality for the canonical Gaussian measure. It generalizes the Gaussian isoperimetric inequality. The equation...

σφαῖρα, sphaîra) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are...

path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called...

(and therefore with the smallest SA:V) is a ball, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with acute-angled spikes...

inequality Geometric measure theory Isoperimetric inequality in higher dimensions Minkowski–Bouligand dimension Federer 1969, p. 273 Krantz & Parks 1999...

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 inequality and the isoperimetric inequality. Let K and L be two n-dimensional convex bodies in n-dimensional Euclidean space Rn. Define a quantity...

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...

In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal...

theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved...

a curve Area – Size of a two-dimensional surface Circumgon – Geometric figure which circumscribes a circle Isoperimetric inequality – Geometric inequality...

edges of an n-dimensional hypercube. For instance, the cube graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has...

restating the problem as a minimization of the Rayleigh quotient. The isoperimetric inequality can be deduced from the Pólya–Szegő inequality with p = 1...

function. The formula plays a decisive role in the modern study of isoperimetric problems. For smooth functions the formula is a result in multivariate...

be calculated using the "Surveyor's formula" (shoelace formula). The isoperimetric inequality states that, for a closed curve of length L (so the region...

algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has...

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...