Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

On the homogeneity of additive mappings

LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH _f :={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH _f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Zur Analytischen Darstellung des Randes Konvexer Körper Durch Stützhyperboloide

In this note, semiaxis-function and vertex-surface of a compact convex subsetK (with interior points) in (n+1)-dimensional Euclidean space are investigated. The semiaxisfunction ofK is defined by associating the half length a of the transverse axisv of any support hyperboloid ofK with the unit vectoru parallel tov, i.e. a is defined as function ofu. The end points of the vectors a(u)·u form the vertex-hypersurface ofK. Most significant result is the continuous differentiability of a (u(t)) as a function of t, where u (t)=1 andu (t) is a continuous differentiable vector function. Proof does not require further assumptions concerningK.     

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation

Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, f ⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x _n} inX is called diagonally stationary for {f ⁿ} if for alln there existsx* _n f ⁿ (x _n) such that x* _{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Sublattices of Lattices of Convex Subsets of Vector Spaces

Let Co(V) be a lattice of convex subsets of a vector space V over a totally ordered division ring . We state that every lattice L can be embedded into Co(V), for some space V over . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional; in this case L also embeds into a finite lower bounded lattice of the form Co(V,)={X | X Co(V)}, for some finite subset of V. This result yields, in particular, a new universal class of finite lower bounded lattices.     

On a class of composite functional equations in a single variable

Summary We prove a general result on continuous functions of the typef: (0, ) (0, ) which satisfy the functional equationf(x[f(x)]^p) = (f(x))^{p + 1}, wherep is an arbitrary fixed real number. Applying this result we determine all continuous solutionsf: [0, ) [0, ) forp > 0, as well as all the continuous solutionsf: for a positive integerp. Forp = 1 this equation is relevant to a division model of population.     

Minty Variational Inequalities,Increase-Along-Rays Property and Optimization<Superscript>1</Superscript>

Let E be a linear space, let K E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^,K ), which can be considered as a generalization of MVI (f ^,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^K ) and the property of f to increase-along-rays starting at x ^*, fIAR (K,x ^*). We prove that the GMVI (f ^,K ) with radially l.s.c. function f has a solution x ^* ker K if and only if fIAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^,K ) has a solution x ^*K, then x ^* is a global minimizer of the problem min f(x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.     

Lower Subdifferentiability and Integration

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that Tf. If is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where f is quasiconvex and is the lower subdifferential ^<. This leads to the introduction of a property that is reminiscent to cyclic monotonicity. We also consider the question of the density of the domains of subdifferential operators.     

Approximation and regular perturbation of optimal control problems via Hamilton-Jacobi theory

We present two convergence theorems for Hamilton-Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. LetT andT _h be the minimal time functions to reach the origin of two control systemsy = f(y, a) andy = f _h (y, a), both locally controllable in the origin, and letK be any compact set of points controllable to the origin. If f _h –f Ch, then |T(x) – T _h (x)| C _K h , for all x K, where is the exponent of Hölder continuity ofT(x).