The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on lR and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in ...

In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R. We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized prob...

Abstract We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave such spaces satisfy several strong continuity properties.

In this paper, we give a new definition of functional Steiner symmetrizations on logconcave functions. Using the functional Steiner symmetrization, we give a new proof of the classical Prékopa-Leindler inequality on log-concave functions.

A probabilistic voting model with voter utility functions that are not necessarily concave is examined. When voters are polarized, there is a convexity threshold of their utility function below which policy convergence is a unique equilibrium, and above which policy divergence is a unique equilibrium. Divergent equilibrium is more likely when voters become more polarized. Social welfare is maxi...

This paper sheds a new light on the split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave function, Bandelt-Dress’ split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics ...

We extend the notion of minimal volume ellipsoid containing a convex body in $$\mathbb {R}^{d}$$ to setting logarithmically concave functions. consider vast class functions whose superlevel sets are concentric ellipsoids. For fixed function from this class, we set all its “affine” positions. any log-concave f on {R}^{d},$$ belonging positions, and find one with integral under condition that it ...

Contributions to the Theory of Optimal Stopping for One–Dimensional Diffusions Savas Dayanik Advisor: Ioannis Karatzas We give a new characterization of excessive functions with respect to arbitrary one–dimensional regular diffusion processes, using the notion of concavity. We show that excessive functions are essentially concave functions, in some generalized sense, and vice–versa. This, in tu...

Abstract In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for class twice differentiable functions, where convexity property target function is not assumed in advance. They represent a refinement case convex/concave functions with numerous applications.

The aim of this paper is to study the behaviour of certain sequence of nonlinear Durrmeyer operators $ND_{n}f$ of the form $$(ND_{n}f)(x)=intlimits_{0}^{1}K_{n}left( x,t,fleft( tright) right) dt,,,0leq xleq 1,,,,,,nin mathbb{N}, $$ acting on bounded functions on an interval $left[ 0,1right] ,$ where $% K_{n}left( x,t,uright) $ satisfies some suitable assumptions. Here we estimate the rate...