2009
DOI: 10.1007/s00526-009-0256-z
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Quantitative isoperimetric inequalities for a class of nonconvex sets

Abstract: Quantitative versions (i.e., taking into account a suitable ``distance'' of a set from being a sphere) of the isoperimetric inequality are obtained, in the spirit of a paper by Fusco, Maggi and Pratelli (Ann. Math. 2008), for a class of not necessarily convex sets called sets with positive reach. Our work is based on geometrical results on sets with positive reach, obtained using methods of both nonsmooth analysis and geometric measure theory

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Cited by 2 publications
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“…It is known (see, e.g., [22,Theorem 4.8 (8)]) that if 4ϕ 0 r < 1 and d(x, K) < r then the metric projection π K (x) = {y ∈ K : |y − x| = d(x, K)} is a singleton, which is Lipschitz with respect to x, and Lip(π K ) = 1 1−4ϕ0r . Moreover, if the diameter of K, diam K, is such that r = 2n n + 1 (ϕ 0 · diam K) 2 < 1 , then the metric projection π K from the convex hull of K, co K into K is a Lipschitz singlevalued map, with Lip(π K ) = 1/(1 − r) (see [18,Proposition 3.1] and [22,Theorem 4.8 (8)]). Actually in this case K is a retract of co K, with a special Lipschitz retraction, namely the metric projection (see also [24]).…”
Section: Appendix Parameterization Of Multifunctionsmentioning
confidence: 98%
“…It is known (see, e.g., [22,Theorem 4.8 (8)]) that if 4ϕ 0 r < 1 and d(x, K) < r then the metric projection π K (x) = {y ∈ K : |y − x| = d(x, K)} is a singleton, which is Lipschitz with respect to x, and Lip(π K ) = 1 1−4ϕ0r . Moreover, if the diameter of K, diam K, is such that r = 2n n + 1 (ϕ 0 · diam K) 2 < 1 , then the metric projection π K from the convex hull of K, co K into K is a Lipschitz singlevalued map, with Lip(π K ) = 1/(1 − r) (see [18,Proposition 3.1] and [22,Theorem 4.8 (8)]). Actually in this case K is a retract of co K, with a special Lipschitz retraction, namely the metric projection (see also [24]).…”
Section: Appendix Parameterization Of Multifunctionsmentioning
confidence: 98%