Let F be a family of sets in R^(d)(always d≥2).A set M■R^(d)is called F-convex,if for any pair of distinct points a,y∈M,there is a set F∈F such that x,y∈F and F■M.We obtain theΓ-convexity,when F consists of F-paths.AΓ-path is the union of both shorter sides of an isosceles right triangle.In this paper we first characterize some I-convex sets,bounded or unbounded,including triangles,regular polygons,subsets of balls,right cylinders and cones,unbounded planar closed convex sets,etc.Then,we investigate the F-starshaped sets,and provide some conditions for a fan,a spherical sector and a right cylinder to be F-starshaped.Finally,we study theΓ-triple-convexity,which is a discrete generalization ofΓ-convexity,and provide characterizations for all the 4-point sets,some 5-point sets and Z^(d)to beΓ-triple-convex.
Given an open bounded subset Ω of ℝ^(n) we consider the eigenvalue problem{Δu-(■u,■V)=-λvu,u>0inΩ,u=0 onδΩ,where V is a given function defined inΩandλV is the relevant eigenvalue.We determine sufficient conditions on V such that ifΩis convex,the solution u is log-concave.We also determine sufficient conditions ensuring that λ_(V),as a function of the setΩ,verifies a convexity inequality with respect to the Minkowski addition of sets.
This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.
Stochastic electricity markets have drawn attention due to fast increase of renewable penetrations.This results in two issues:one is to reduce uplift payments arising from non-convexity under renewable uncertainties,and the other one is to allocate reserve costs based on renewable uncertainties.To resolve the first issue,a convex hull pricing method for stochastic electricity markets is proposed.The dual variables of system-wide constraints in a chance-constrained unit commitment model are shown to reduce expected uplift payments,together with developing a linear program to efficiently calculate such prices.To resolve the second issue,an allocation method is proposed to allocate reserve costs to each renewable power plant by explicitly investigating how renewable uncertainties of each renewable power plant affect reserve costs.The proposed methods are validated in a 24-period 3-unit test example and a 24-period 48-unit utility example.
