On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.     

Even valuations on convex bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.

     

Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne

We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted considerable attention in convex analysis. Our results imply that the minimizers of proximal-average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions. When both convex functions in the proximal average are indicator functions of convex sets, least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings. This provides a partial answer to a question posed by C. Byrne. A novelty of our approach is to use the notion of resolvent average and proximal average.     

一个周期函数不等式及其应用

本文研究一类特殊的周期函数,利用Fourier级数的方法,获得了关于这类周期函数的一个积分不等式.此函数积分不等式等价于著名的关于平面两凸集混合面积的Minkowski不等式.     

On Some Generalized Polyhedral Convex Constructions

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.     

Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization,reverse convex constraints,DC-programming,and Lipschitzian optimization

A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.     

Convex and strongly convex fuzzy sets

Properties of the support and of the core of convex and strongly convex fuzzy sets are considered. The convex and strongly convex fuzzy sets in the real line are characterized by means of the piece-wise monotonic functions.     

New Combinations of Convex Sets

The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.     

Wirtinger不等式的一个几何应用

本文研究了著名的Minkowski混合面积不等式.利用平面凸集的支持函数的性质和分析中著名的Wirtinger不等式,得到了Minkowski混合面积不等式的一个简化证明.     

A refinement of the Bernštein-Kušnirenko estimate

A theorem of Kušnirenko and Bernštein (also known as the BKK theorem) shows that the number of isolated solutions in a torus to a system of polynomial equations is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

Convex envelopes generated from finitely many compact convex sets

We consider the problem of constructing the convex envelope of a lower semi-continuous function defined over a compact convex set. We formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets. In particular, we consider nonnegative functions that are products of convex and component-wise concave functions and derive closed-form expressions for the convex envelopes of a wide class of such functions. Several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms

In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.     

Condition numbers and error bounds in convex programming

After a brief survey on condition numbers for linear systems of equalities, we analyse error bounds for convex functions and convex sets. The canonical representation of a convex set is defined. Other representations of a convex set by a convex function are compared with the canonical representation. Then, condition numbers are introduced for convex sets and their convex representations.     

E-凸函数的若干特征

讨论了一类广义的凸集和凸函数:E-凸集和E-凸函数的若干性质,并给出E-凸函数的一个判别准则．     

Analysis to Neyman-Pearson classification with convex loss function

Neyman-Pearson classification has been studied in several articles before.But they all proceeded in the classes of indicator functions with indicator function as the loss function,which make the calculation to be difficult.This paper investigates NeymanPearson classification with convex loss function in the arbitrary class of real measurable functions.A general condition is given under which Neyman-Pearson classification with convex loss function has the same classifier as that with indicator loss function.We give analysis to NP-ERM with convex loss function and prove it's performance guarantees.An example of complexity penalty pair about convex loss function risk in terms of Rademacher averages is studied,which produces a tight PAC bound of the NP-ERM with convex loss function.     

Mixed Measures of Convex Cylinders and Quermass Densities of Boolean Models

Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper, we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them. After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing classical formulas by Davy and recent results by Spiess and Spodarev.