A differentiable structure for metric measure spaces

Keith, Stephen

doi:10.1016/s0001-8708(03)00089-6

Cited by 106 publications

(151 citation statements)

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“…Metric spaces equipped with doubling measures that support Poincaré inequalities (also known as PI spaces) are ideal environments for first-order analysis and differential geometry [20], [10], [21], [24], [26]. Extending the scope of this theory by verifying Poincaré inequalities on new classes of spaces is a problem of high interest and relevance.…”

Section: Introductionmentioning

confidence: 99%

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

2013

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Abstract. A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.

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Section: Introductionmentioning

confidence: 99%

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

2013

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“…An influential work in this direction has been [22], where conditions were given for the existence of a differentiable structure on metric measure spaces, based on Lipschitz maps, which also have the doubling property and admit a Poincaré inequality. Among the numerous works that clarified, elaborated and generalized [22], we could point out [23][24][25][26]. In these works, the differentiable structure appears naturally as a result of the more "primitive" assumptions stated above and presented in [22].…”

Section: Reflexive and Super-reflexive Spacesmentioning

confidence: 99%

Convexity and the Euclidean Metric of Space-Time

Kalogeropoulos

2017

Universe

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Abstract:We address the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

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“…Moreover, (2.35) holds with K = 1; (Keith): [Kei02,Kei04]; if (X, µ) is a doubling metric measure space which satisfies the Lip-lip inequality (2.35), then (X, µ) is a differentiability space whose analytic dimension is bounded by an expression that depends only on C µ and K. Moreover [Kei03], the Poincaré inequality is stable under measured Gromov-Hausdorff convergence provided all the relevant constants are uniformly bounded; for example, blow-ups of PI-spaces are PI-spaces (with the same PI-exponent); (Bate-Speight): [BS13]; if (X, µ) is a differentiability space then µ is asymptotically doubling in the sense that for µ-a.e. x there are (C x , r x ) ∈ (0, ∞) 2 such that:…”

Section: Definition 222 (Speed Of Alberti Representations)mentioning

confidence: 99%

“…In his PhD thesis, Keith [Kei02,Kei04] introduced a new analytic condition, the Lip-lip inequality, and proved that doubling metric measure spaces (X, µ) satisfying it are differentiability spaces. It seems that the idea of "generalizing" Cheeger's Differentiation Theorem using a Lip-lip inequality stems from the fact that in PIspaces Cheeger had proven a Lip-lip equality.…”

Section: Introductionmentioning

confidence: 99%