Some connections between isoperimetric and Sobolev-type inequalities

Bobkov, Serguei G.; Houdré, Christian

doi:10.1090/memo/0616

Cited by 112 publications

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“…Borell [5] and Sudakov with Tsirelson [12] proved that in this case half-spaces {x : x, u ≥ λ} are extremal. As Bobkov and Houdré proved in [4], on the real line this result can be generalized into the case of an arbitrary symmetric log-concave measure. Bobkov [3] also studied the isoperimetric problem in the product metric space (X n , d sup ) equipped with a product probability measure, where d sup (x, y) := sup i≤n d(x i , y i ).…”

Section: Introduction and Main Resultsmentioning

confidence: 64%

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Strzelecka

2017

Positivity

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Section: Introduction and Main Resultsmentioning

confidence: 64%

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Strzelecka

2017

Positivity

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“…Recall the two definitions we have discussed so far, γ + (A) from (1) and surf(A) from (2). There is also a third natural definition, namely the integral of the Gaussian density over the boundary of A.…”

Section: A Definitions Of Gaussian Surface Areamentioning

confidence: 99%

Gaussian noise sensitivity and Fourier tails

2018

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“…We shall apply a standard scheme of proof which originates from a paper of Federer and Fleming [9, Remark 6.6] and which later found frequent applications to similar questions (cf., for instance, [16], [5]). …”

Section: Examplementioning

confidence: 99%

Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds

Miklyukov

Вуоринен

1999

Proc. Amer. Math. Soc.

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Abstract. We prove that for every Riemannian manifold X with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class W 1,p 0 (X ). We also study manifolds satisfying Hardy's inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold. Main resultsLet X be a connected noncompact Riemannian C 2 -manifold without boundary. We denote by ρ(x , x ) the distance between two points x , x ∈ X . For an arbitrary point x ∈ X we set ε(x) = infwhere the infimum is taken over all sequences of points (y k ) ⊂ X , which do not have points of accumulation in X . Let α(t), β(t) : [0, ∞) → [0, ∞) be positive continuous functions. We fix constants p, q such that 1 < p ≤ q < ∞. Let Ω ⊂ X be an arbitrary domain, and let ∂Ω be its boundary. We consider a weighted volumeHere, dv is the element of volume on the manifold X . Below we shall assume thatWe denote a weighted areawhere dH n−1 is the element of the (n − 1)-dimensional Hausdorff measure on ∂Ω. We consider isoperimetric profiles of the Riemannian manifold X with weight functions α and β. An isoperimetric profile of the manifold X is the function θ X : [0, I X ) → R + , θ(0) = 0, (1.1)

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Some connections between isoperimetric and Sobolev-type inequalities

Cited by 112 publications

References 0 publications

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Gaussian noise sensitivity and Fourier tails

Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds

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