2 results
On convex sets that minimize the averagedistance
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 18 / Issue 4 / October 2012
- Published online by Cambridge University Press:
- 16 January 2012, pp. 1049-1072
- Print publication:
- October 2012
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In this paper we study the compact and convex setsK ⊆ Ω ⊆ ℝ2 that minimize
\begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*}for some constants λ1 andλ2, that could possibly be zero. We compute in particularthe second order derivative of the functional and use it to exclude smooth points ofpositive curvature for the problem with volume constraint. The problem with perimeterconstraint behaves differently since polygons are never minimizers. Finally using a purelygeometrical argument from Tilli [J. Convex Anal. 17 (2010)583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeterand volume constraints are considered.
Generalized regular variation of second order
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- Journal:
- Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 61 / Issue 3 / December 1996
- Published online by Cambridge University Press:
- 09 April 2009, pp. 381-395
- Print publication:
- December 1996
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Assume that for a measurable funcion f on (0, ∞) there exist a positive auxiliary function a(t) and some γ ∈ R such that
. Then f is said to be of generalized regular variation. In order to control the asymptotic behaviour of certain estimators for distributions in extreme value theory we are led to study regular variation of second order, that is, we assume that 
exists non-trivially with a second auxiliary function a1(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corresponding relation for the inverse function of a monotone f with the stated property. Finally, we present an Abel-Tauber theorem relating these functions and their Laplace transforms.
. Then 
exists non-trivially with a second auxiliary function a