The alternating algorithm in a uniformly convex and uniformly smooth Banach space

Let X   be a uniformly convex and uniformly smooth Banach space. Assume that the _Mi$M_i$, i=1,…,r$i=1,\dots ,r$, are closed linear subspaces of X  , _PMi$P_{M_i}$ is the best approximation operator to the linear subspace _Mi$M_i$, and M:=_M1+?+_Mr$M:=M_1+?+M_r$. We prove that if M is closed, then the alternating algorithm given by repeated iterations of

(I−_PMr)(I−_PMr−1)?(I−_PM1)$(I-P_{M_r})(I-P_{M_{r-1}})?(I-P_{M_1})$

applied to any x∈X$x\in X$ converges to x−_PMx$x-P_{M}x$, where _PM$P_M$ is the best approximation operator to the linear subspace M  . This result, in the case r=2$r=2$, was proven in Deutsch [4].     

An isoperimetric inequality in the optimization of circular rings under normal pressure

The problem of determining the optimal cross section of a circular ring so as to maximize the buckling pressure under a given total volume is formulated and solved. An isoperimetric inequality is proved: Among all the circular rings of given mass and radius, the ring with constant bending rigidity along the arc length has the largest critical buckling pressure.     

A two-dimensional inequality and uniformly continuous retractions

We prove a new inequality valid in any two-dimensional normed space. As an application, it is shown that the identity mapping on the unit ball of an infinite-dimensional uniformly convex Banach space is the mean of n uniformly continuous retractions from the unit ball onto the unit sphere, for every n?3. This last result allows us to study the extremal structure of uniformly continuous function spaces valued in an infinite-dimensional uniformly convex Banach space.     

On the stability of the polygonal isoperimetric inequality

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices.     

An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ? (approximately 1.71579) for which any simple closed path α in the universal cover of R²?Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥?A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R²?Z².     

Remarks on the structure of the fixed-point sets of uniformly lipschitzian mappings in uniformly convex Banach spaces

It is shown, by asymptotic center techniques, that the set of fixed points of any uniformly k-lipschitzian mapping in a uniformly convex Banach space is a retract of the domain when k is less than a constant bigger than the constant from the paper [K. Goebel, W.A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973) 135-140]. Our result improves a recently result presented in [E. S?d?ak, A. Wi?nicki, On the structure of fixed-point sets of uniformly lipschitzian mappings, Topol. Methods Nonlinear Anal. 30 (2007) 345-350].     

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ?C along ?Σ∩?C and ?Σ ～ ?C is radially connected from a point p ∈ ?Σ∩?C. We introduce a modified volume M_p(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

An isoperimetric inequality for convex polygons and convex sets with the same symmetrals

Suppose that two distinct plane convex bodies have the same Steiner symmetrals about a finite number n of given lines. Then we obtain an upper bound for the measure of their symmetric difference. The bound is attained if, and only if, the directions of the lines are equally spaced and the bodies are two regular concentric polygons, with n sides, each obtained from the other by rotation through an angle /n. This result follows from a new isoperimetric inequality for convex polygons.     

A subclass of parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. A function f in A satisfying the normalization is said to be in the class SP_n if Dⁿf is a parabolic starlike function, where Dⁿ is a notation of the Salagean operator. In this paper, several basic properties and characteristics of the class SP_n are investigated. These include subordination, convolution properties, class-preserving integral operators, and Fekete-Szegö problems.     

A sharp isoperimetric bound for convex bodies

We consider the problem of lower bounding the Minkowski content of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp. Supported in part by VIGRE grants at Yale University and the Georgia Institute of Technology.     

Absolutely convex,uniformly starlike and uniformly convex harmonic mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disc. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi-type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.     

An isoperimetric comparison theorem for Schwarzschild space and other manifolds

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric -spheres of a spherically symmetric -manifold are isoperimetric hypersurfaces, meaning that they minimize -dimensional area among hypersurfaces enclosing the same -volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual -dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.