On the isoperimetric problem for radial log-convex densities
نویسندگان
چکیده
منابع مشابه
On the Isoperimetric Problem for Radial Log-convex Densities
E e as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0 > 0; this affirmatively answers conjecture [RCBM, Conjecture 3.12] for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurca...
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Let μ = ρdx be a Borel measure on Rd. A Borel set A ⊂ R is a solution of the isoperimetric problem if for any B ⊂ R satisfying μ(A) = μ(B) one has μ(∂A) ≤ μ(∂B), where μ(∂A) = ∫ ∂A ρ dHd−1 is the corresponding surface measure. There exists only a small number of examples where the isoperimetric problem has an exact solution. The most important case is given by Lebesgue measure λ on R, the solut...
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15 صفحه اولAn isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies
We prove an isoperimetric inequality for uniformly log-concave measures and for the uniform measure on a uniformly convex body. These inequalities imply the log-Sobolev inequalities proved by Bobkov and Ledoux [12] and Bobkov and Zegarlinski [13]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov and Milman [22].
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formulation is useful in some areas such as discrete geometry, probability and Banach space theory; see for instance [36] and [49]. The Isoperimetric profile Iμ = I(ds2,μ) of (M, μ) is defined, as above, by Iμ(t) = inf {μ + (R) : μ(R) = t}, 0 ≤ t ≤ μ(M). If we change the metric or the measure by a positive factor a, it follows easily that Iaμ(t) = aIμ( t a ) and I(a2ds2,μ) = 1 a I(ds2,μ). Among...
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ژورنال
عنوان ژورنال: Calculus of Variations and Partial Differential Equations
سال: 2012
ISSN: 0944-2669,1432-0835
DOI: 10.1007/s00526-012-0557-5