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In Groeneboom (1988) a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of convex polygon was derived. In the unpublished preprint Nagaev and Khamdamov (1991) (in Russian) a central limit result for the joint distribution of the number of vertices and the remaining area is given, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of the number of vertices and the remaining area in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Nagaev (1995) and Cabo and Groeneboom (1994)., 18 pages, 4 figures, To appear in the Advances in Applied Probability

We study "heavy" A?-dimensional surfaces suspended from some prescribed (n — 1) -dimensional boundary data. This leads to a mean curvature type equation with a non-monotone right hand side. We show that the equation has no solution if the boundary data are too small, and, using a fixed point argument, that the problem always has a smooth solution for sufficiently large boundary data.

This work concerns boundary value problems for a class of nonlinear equations modeled on the physical equations for a capillary free surface in a gravitational field. The results consist principally of estimates for the height of a solution in an exterior domain. Structure conditions reflecting the nonlinearity of the mean curvature operator are imposed on a class of symmetric variational operators in terms of the Legendre transform of the variational integrand. Estimates are found for the boundary height of a rotationally symmetric solution in the exterior of a ball of radius R. These estimates, which are valid for any R9 are shown to be asymptotically exact as R tends to zero or infinity. This provides a proof of the asymptotic behavior of the boundary height which previously has been derived by a formal perturbation method. An asymptotic characterization of the solution in a neighborhood of the boundary is also given. For a general domain estimates are obtained from a maximum principle due to Finn in which the symmetric solutions serve as comparison functions. l Preliminaries* We begin by formulating the "standard" capillarity problem on an exterior domain n dimensional space; the physical case is n = 2. Let Ω c Rn be an exterior domain whose boundary, Σ, is a compact C1 hypersurface. When the generalized (vertical) cylinder Σ x R is immersed in an infinite reservoir of fluid, the action of capillarity gives rise to a free surface of static equilibrium in the outside of the cylinder. Let the height of this capillary free surface, assumed nonparametric over Ω, be given by the scalar function u(x)f x — (xlf - , xn) eΩ. Physical principles assert that the equilibrium free surface minimizes the functional

We consider the usual capillarity problem with the additional requirement that the capillary surface lies above some obstacle. This involves a variational inequality instead of a boundary value problem. We prove existence of a solution to the variational inequality and study the boundary regularity. In particular, global C 11 -regularity is shown for a wider class of variational inequalities with conormal boundary condition.

On considere le probleme de Dirichlet pour l'equation d'une surface minimale sur un domaine borne de R n avec une courbure moyenne non negative. On donne un module de continuite de la solution en termes de module de continuite des valeurs frontieres

A new global approach for the variational equations of fluids is given. The concept of prefluid is introduced, together with its variational equations and examples.

The inequality by Polya and Schiffer considered in this paper is concerned with the sums of the n first reciprocal eigenvalues of the problem Au + Tui = 0 in G, u = 0 on SG. First we extend this inequality to the problem of an inhomogeneous membrane Au + Apu = 0 in G, u = 0 on SG. Then we prove a sharper form of it for a class of homogeneous membranes with partially free boundary. The proofs are based on a variational characterization for the eigenvalues and use conformal mapping and transplantation arguments. 'The work was supported by NSF Grant GU-2056. EXTENSIONS OF AN INEQUALITY BY POLYA AND SCHIFFER FOR VIBRATING MEMBRANES Catherine Bandle INTRODUCTION. The inequality by Polya and Schiffer considered in this paper is concerned with the eigenvalue problem Atp + A . The Gaussian curvature has the form K =(~A In p)/2p r &2 S2 £_ = — 5 — + — 7 . We shall assume that the in equality L Z Bx ay J " K 0. The a a conformal radius of the point a with respect to G is then defined as r (G) = 1/f T (a) [9, p. 16] . We set a a (2) R (G) = < a. (G) if Ko ^ 0 \/p(a) 'ra(G) if K Q = 0 Example; If G is a circle with the radius r 3 the center in the origin and p(z) = g(z), then R (G) = r . w (z) = R (G)f (z) maps G onto the circle {w; Iwl < R (G)}. a a a. a and z (w) denotes its inverse. We shall denote the circle a {w; |wI < e} by C . R a( ) h a s bean chosen in such a way that (3) Jj g(w)dudv = JJ p(z)dxdy -f o(c). c Since JJ g(w)dudv = < ^47rcc + o(€ ) if K Q £ O € W e 2 if

Let Γ be a Jordan curve in R and F(z) = (u(z)9 v(z), w(z)): {\z\ ^ 1} -» R be a solution of Plateau's problem for Γ, where z = x + iy are isothermal parameters. Then u,v,w are harmonic in {\z\ * for | z \ 1, where ω*(t) is a certain modulus of continuity. Once again ω* depends only on Γ.

On etudie des surfaces minimales completes bornees par des lignes dans R 3 . On demontre des theoremes de Bernstein pour de telles surfaces