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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 area of a hypersurface that divides M into two disjoint pieces. In 1971, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on ...
In the context of metric measure spaces, the definition of a Poincaré inequality is slightly different.One definition is: a metric measure space supports a (q,p)-Poincare inequality for some , < if there are constants C and λ ≥ 1 so that for each ball B in the space, ‖ ‖ () ‖ ‖ ().
In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling.
Proof [2]. Since + =, =. A graph = on the -plane is thus also a graph =. From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines =, =, =, =, and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality ...
gives the inequality. In the special case of n = 1, the Nash inequality can be extended to the L p case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (Brezis 2011, Comments on Chapter 8). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds
To obtain the actual crossing number inequality, we now use a probabilistic argument. We let 0 < p < 1 be a probability parameter to be chosen later, and construct a random subgraph H of G by allowing each vertex of G to lie in H independently with probability p, and allowing an edge of G to lie in H if and only if its two vertices were chosen ...
The resulting graph is the Turán graph (,). Turán's theorem states that the Turán graph has the largest number of edges among all K r+1-free n-vertex graphs. Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. [1]
Shearer's inequality or also Shearer's lemma, in mathematics, is an inequality in information theory relating the entropy of a set of variables to the entropies of a collection of subsets. It is named for mathematician James B. Shearer .
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