A mass transportation approach to quantitative isoperimetric inequalities

Figalli, Alessio; Maggi, Francesco; Pratelli, Aldo

doi:10.1007/s00222-010-0261-z

Cited by 301 publications

(342 citation statements)

References 36 publications

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“…The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42,72,102], as well as applications in physics [38,75,97], statistics [88], engineering [15,16,57,86,107], atmospheric modeling [33][34][35]87], and economics [24,25,27,41,49]. The second is a problem in functional analysis, at the junction between measure theory and convex geometry.…”

Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

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Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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“…Surprisingly enough, it seems that these results are new, although similar ones can be deduced from the much more general [FMP09] (but with a non-optimal constant) and the Bonnesen-like inequalities of [PWZ93] (but only when A is convex).…”

Section: Proposition 1 Let a Be A Domain Of The Lmentioning

confidence: 57%

Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

Kloeckner

2010

Proc. Amer. Math. Soc.

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Abstract. An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the L 1 case has drawn much less attention.In this note we prove two quantitative isoperimetric inequalities in the L 1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch. Statement of the resultsWe consider the plane R 2 endowed with the L 1 metricWe denote the boundary of a set by ∂. The notation | · | shall be used to denote the size of an object, whatever its nature. If A is a measurable plane set, then |A| is its Lebesgue measure, also called its area; if v is a vector, |v| is its L 1 norm; if γ is a curve, |γ| is its L 1 length, namelywhere the supremum is over all increasing sequences (t 1 , t 2 , . . . , t n ) taking values in the definition interval of γ. A curve is said to be (L 1 -)rectifiable if its length is finite. For example, a curve defined on a segment whose coordinate functions are both monotonic is rectifiable, and its length is the distance between its endpoints. See e.g. [AT04] for more information on the length of curves in a metric space.By a domain of the plane, we mean the closure of the bounded component of a Jordan curve. In particular, domains are compact and connected. All rectangles and squares considered are assumed to have their sides parallel to the coordinate

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“…Fusco, Maggi and Pratelli [6] gave a sharp estimate for the Fraenkel asymmetry (see (1.1)) of a set E in terms of its isoperimetric deficit, proving a conjecture by Hall [9]. Very recently, an anisotropic version of the result in [6] was established by Figalli, Maggi and Pratelli in [3]. More precisely, given a convex set E , 0 < |E| < +∞, containing the origin, and an open set F , 0 < |F | < +∞, with smooth boundary ∂F oriented by its unit outer normal ν F , the anisotropic perimeter of F is defined as…”

Section: Introductionmentioning

confidence: 65%

“…In the last few years, the so-called quantitative isoperimetric inequalities have attracted a great interest (see for example [6,3] and the references therein). In order to describe these results let us introduce, for any Borel set E in R n , with 0 < |E| < +∞, the isoperimetric deficit of E δ(E) = P (E)…”

Section: Introductionmentioning

confidence: 99%