Contributions to max-min convex geometry. I: Segments

We give some contributions to the theory of “max-min convex geometry”, that is, convex geometry in the semimodule over the max-min semiring R_max,min=R∪{-∞,+∞}. We introduce “elementary segments” that generalize from n=2 the horizontal, vertical or oblique segments contained in the main bisector of . We show that every segment in is a concatenation of a finite number of elementary subsegments (at most 2n-1, respectively at most 2n-2, in the case of comparable, respectively, incomparable, endpoints x,y). In this first part we study “max-min segments”, and in the subsequent second part (submitted) we study “max-min semispaces” and some of their relations to “max-min convex sets”.     

Convex regularization of local volatility models from option prices: Convergence analysis and rates

We study a convex regularization of the local volatility surface identification problem for the Black-Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid-ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero.We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

The structure of max-min hyperplanes

In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form

a₁⊗x₁⊕…⊕a_n⊗x_n⊕a_n+1=b₁⊗x₁⊕…⊕b_n⊗x_n⊕b_n+1,     

Complément de Schur et sous-différentiel du second ordre d'une fonction convexe

Summary The Schur complement relative to the linear mappingA of a functionf is denotedAf and defined as the image off underA. In this paper we give some estimates for the second-order differential ofAf whenf is either a partially quadratic convex function or aC ² convex function with a nonsingular second-order differential. We then consider an arbitrary convex functionf and study the second-order differentiability ofAf in a more general sense.

     

Two-point symmetrization and convexity

We prove a conjecture of R. Schneider: the spherical caps are the only spherically convex bodies of the sphere which remain spherically convex after any two-point symmetrization. More generally, we study the relationships between convexity and two-point symmetrization in the Euclidean space and on the sphere. Received: 4 April 2003     

Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

 We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of convex polytopes as the number of facets tends to infinity. Received 16 July 2001     

On the volume of convex hulls of sets on spheres

We prove that for a measurable subset of S ^n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R ² and R ³.Partially supported by KBN (Poland), Grant No. 2 1094 91 01.     

Immobilization of smooth convex figures

We establish a curvature criterion to decide whether three points immobilize a plane convex figure with smooth boundary. Then we use it to prove in the affirmative the convex case of Kuperberg's Conjecture. Namely, we prove that any convex figure with smooth boundary, different from a circular disk, can be immobilized with three points.     

Sylvester’s Problem for Symmetric Convex Bodies and Related Problems

We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.     

Optimizing Area and Perimeter of Convex Sets for Fixed Circumradius and Inradius

 In this paper two problems posed by Santaló are solved: we determine the planar convex sets which have maximum and minimum area or perimeter when the circumradius and the inradius are given, obtaining complete systems of inequalities for the cases (A, R, r) and (p, R, r). This work is supported in part by Dirección General de Investigación (MCYT) BFM2001-2871, and by OTKA grants No 31984 and 30012 Received October 15, 2001; revised January 29, 2002     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

Max-min controllability in pursuit games with norm-bounded controls

In this article, we consider max-min controllability in linear pursuit games with norm-bounded controls. Our approach is based on the separation theorem of disjoint compact convex sets in Euclidean space. Necessary and sufficient conditions for max-min controllability are given in terms of an explicit relative controllability expression which, in a sense, is a comparison between the control capabilities of the competing parties. Minimal time and optimal norm problems are investigated.     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.