Related papers: Compensated Convex Transforms and Geometric Singul…

We study local asymptotic normality of M-estimates of convex minimization in an infinite dimensional parameter space. The objective function of M-estimates is not necessary differentiable and is possibly subject to convex constraints. In…

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev…

We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean $n$-spaces, $n\geq 3$. The inner and outer distortion functionals are lower…

If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that…

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, we then…

We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence…

In this paper we show how the superquadratic functions can be used as a tool for researching other types of convex functions like $\phi $-convexity, strong-convexity and uniform convexity. We show how to use inequalities satisfied by…

In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly convex…

If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\dist^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance…

We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…

This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain…

For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…

In this paper, some new inequalities of the Hermite-Hadamard type for h- convex functions whose modulus of the derivatives are h-convex and applications for special means are given.

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this…

A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex…

Typically, the sequence of points generated by an optimization algorithm may have multiple limit points. Under convexity assumptions, however, (sub)gradient methods are known to generate a convergent sequence of points. In this paper, we…

This paper provides an unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set valued functions.

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…

We consider the problem of finding $\lambda\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{\lambda + F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic,…

Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…