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A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x … Ax of the Euclidean case must be replaced by suitable group isomorphisms x … TA(x), whose differential preserves the first layer of the stratification of Lie(G). [ABSTRACT FROM AUTHOR]

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory. [ABSTRACT FROM AUTHOR]

We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly distributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behaviour. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory. [ABSTRACT FROM AUTHOR]

In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of $mathbb {R}^{3}$ which sends convex sets to convex ones is affine. It can be remarked also that a $mathcal {C}^{2}$-diffeomorphism $F:Uto U^{'}$ between two domains in $mathbb {C}^{n}$, $nge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. par In this paper we are interested in the self-mappings of $mathbb {C}^{n}$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {em A } $mathcal {C}^{2}$-{em diffeomorphism} $F:U'to U$ {em (where } $U', Usubset mathbb {C}^{n}$ {em are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map} $Phi := F^{-1}$ {em is weakly pluriharmonic, } i.e., {em if it satisfies some nice second order PDE very close to} $partial bar {partial }Phi = 0$. In fact all pluriharmonic $Phi $'s do satisfy this equation, but there are also other solutions. [ABSTRACT FROM AUTHOR]

A real valued function $f$ defined on a real interval $I$ is called $(\varepsilon,\delta)$-convex if it satisfies \[ f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)|x-y| + \delta \quad \text{for} x,y\in I, t\in[0,1]. \] The main results of the paper offer various characterizations for $(\varepsilon,\delta)$-convexity. One of the main results states that $f$ is $(\varepsilon,\delta)$-convex for some positive $\varepsilon$ and $\delta$ if and only if $f$ can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case $\varepsilon=0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called $\delta$-convexity. [ABSTRACT FROM AUTHOR]

Let $\Phi (t)$ and $\Psi (t)$ be nonnegative convex functions, and let $\varphi $ and $\psi $ be the right continuous derivatives of $\Phi $ and $\Psi ,$ respectively. In this paper, we prove the equivalence of the following three conditions: (i) $\|f^{*}\| _\Phi \leq c\|f\|_\Psi ,$ (ii) $L^\Psi \subseteq $ $H^\Phi $ and (iii) $\int_{s_0}^t\frac{\varphi (s)}sds\leq c\psi (ct), \forall t>s_0,$ where $L^\Psi $ and $H^\Phi $ are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities. [ABSTRACT FROM AUTHOR]

This paper presents a necessary and sufficient condition on the convex function $f$ in order that continuous solutions to \[\hbox {minimize} \int _{\Omega } f(\|\nabla u(x)\|) dx \hbox { on } u^{0} + W^{1,1}_{0}(\Omega )\] satisfy a Strong Maximum Principle on any open connected $\Omega $. [ABSTRACT FROM AUTHOR]

Let $X$ be a real, reflexive, locally uniformly convex Banach space with $X^{*}$ locally uniformly convex. ~Let $T:X\supset D(T)\to 2^{X^{*}}$ be a maximal monotone operator and $G\subset X$ open and bounded. ~Assume that $M\subset X^{*}$ is pathwise connected and such that $T(D(T)\cap G)\cap M \not = \emptyset $ and $ \overline{T(D(T)\cap \partial G)}\cap M = \emptyset .$ ~Then $M\subset T(D(T)\cap G).$ ~If, moreover, $T$ is of type ($S$) on $\partial G,$ then $ \overline{T(D(T)\cap \partial G)}$ may be replaced above by $T(D(T)\cap \partial G).$ The significance of this result lies in the fact that it holds for multi-valued mappings $T$ which do not have to satisfy $\text{int}D(T) \not = \emptyset .$ ~It has also been used in this paper in order to establish a general ``invariance of domain'' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. ~No degree theory has been used. ~In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set $G$) of a completely continuous perturbation $T+C$ of a maximal monotone operator $T$ such that $T+C$ is locally monotone on $G.$ [ABSTRACT FROM AUTHOR]

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into C2. We show that the polynomially convex hull of Rudin's Klein bottle contains an open subset of C2. We also describe another totally real Klein bottle in C2 whose hull has topological dimension equal to three. [ABSTRACT FROM AUTHOR]

Let Φ : [0,∞) → R be a continuous convex function with Φ(0) = 0. We prove that Φ(∥f∥1/ωN∥f∥ ∞)⩽ = 1 ∫RN/ωN∥f∥∞ RN ∣f(x)∣Φ(∣x∣N)dx for every f ∈ L1(RN) ∩ L∞(RN), f ≠ 0, where N is the measure of the unit ball of RN. This can be used to obtain lower or upper bounds for weighted integrals ∫RN ∣f(x)∣η(∣x∣)dx in terms of the L1 and L∞ norms of f, which are often much sharper than crude estimates that may be obtained, if at all, by a visual inspection of the integrand. The basic inequality is essentially independent of Jensen's inequality, but it is closely related to Steffensen's inequality. [ABSTRACT FROM AUTHOR]