ON A GLOBALIZATION PROPERTY

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1 APPLICATIONES MATHEMATICAE 22,1 (1993), pp S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X. A function φ Φ is called a local Φ- subgradient of a function f : X R at a point x 0 if there is a neighbourhood U of x 0 such that f(x) f(x 0 ) φ(x) φ(x 0 ) for all x U. A function φ Φ is called a global Φ-subgradient of f at x 0 if the inequality holds for all x X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies the existence of a global Φ-subgradient of f at each point (globalization property), (b) when each local Φ-subgradient can be extended to a global Φ-subgradient (strong globalization property). Let (X, τ) be a topological space. Let f be a real-valued function defined on X. Let Φ be a class of real-valued functions defined on X. We say that the function f is Φ-convex if it can be represented as a supremum of functions belonging to Φ. A function φ Φ is called a local Φ-subgradient of the function f at a point x 0 if there is a neighbourhood U of x 0 such that for all x U, (1) f(x) f(x 0 ) φ(x) φ(x 0 ). A function φ Φ is called a global Φ-subgradient (briefly, Φ-subgradient) of f at x 0 if (1) holds for all x X. It is easy to show that the fact that f has a local Φ-subgradient at each point does not imply that f has a Φ-subgradient at each point, nor even that f is Φ-convex Mathematics Subject Classification: 52A01, 52A99. Key words and phrases: globalization property, Φ-subgradients. The paper is partially supported by the Polish Committee for Scientific Research under grant no

2 70 S. Rolewicz Example. Let X = R. Let Φ denote the class of all quadratic functions. Let f(x) = x 3. Then f is not bounded from below by any function φ Φ. On the other hand, it has a local Φ-subgradient at each point. It is interesting, however, that there are classes Φ such that the existence of a local Φ-subgradient of a function f at each point x 0 X implies the existence of a global Φ-subgradient of f at each point. We then say that Φ has the globalization property. If each local Φ-subgradient can be extended to a global one we say that Φ has the strong globalization property. If the existence of a local Φ-subgradient of a bounded function f at each point x 0 X implies the existence of a global Φ-subgradient of f at each point we say that Φ has the bounded globalization property. Let A X. We say that the set A has the Φ-globalization property (strong Φ-globalization property, bounded Φ-globalization property) if the family Φ restricted to A has the globalization property (resp. strong globalization property, bounded globalization property). In particular, if X is a linear topological space and Φ is the class of continuous linear functionals on X then a set A with the Φ-globalization property will be said to have the (strong, bounded) linear globalization property or briefly the (strong, bounded) globalization property. Proposition 1. Let (X, τ) be a linear topological space. Then X has the strong linear globalization property. P r o o f. We begin with the one-dimensional space X = R. Recall that a function f defined on the real line is convex if and only if (2) lim sup t t 0 +0 f(t) f(t 0 ) t t 0 f(t) f(t 0 ) lim inf. t t 0 0 t t 0 The existence of a local linear subgradient of f at each point implies (2). Thus f is convex. For arbitrary dimension we simply observe that the restriction of f to any one-dimensional subspace is convex. This implies that f is convex. Therefore each local linear subgradient is a (global) linear subgradient. The same considerations give Proposition 2. A convex set in a linear topological space has the strong linear globalization property. It is interesting to know which families of linear functionals have the bounded globalization property. Proposition 3. Let X be the unit sphere in a Banach space (Y, ), and X = {x Y : x = 1}. Let Φ be the family of continuous linear functionals restricted to X. Then Φ has the bounded globalization property.

3 On a globalization property 71 P r o o f. Let f be a bounded function defined on X and having a local Φ-subgradient at each point x 0 X. Let a R be chosen so that f 1 (x) = f(x) a 0 for all x X. We show that f 1 has a Φ-subgradient at each point of X, which automatically implies that so does f. We extend f 1 to the whole space Y putting { f 2 (x) = x f1 (x/ x ) for x 0, 0 for x = 0. It is easy to see that f 2 has local Φ-subgradient 0 at 0, because inf x X (f 2 (x) inf{φ(x) : φ Φ}) 0. Take any point x 0 0. The function f 1 (x/ x ) has a local Φ-subgradient φ 1 at x 0 / x 0. Observe that f 1 (x 0 / x 0 ) φ 1 (x 0 / x 0 ) 0. Let φ 2 denote a functional of norm one supporting X at x 0 / x 0, i.e. such that φ 2 (x 0 / x 0 ) = 1. Observe that the functional φ(x) = φ 1 (x) + bφ 2 (x) is a local linear subgradient of f 1 at x 0 / x 0 for all b 0. If b = f 1 (x 0 / x 0 ) φ 1 (x 0 / x 0 ) then φ(x 0 / x 0 ) = f 1 (x 0 / x 0 ) and f 1 (x) φ(x) in some neighbourhood V of x 0 / x 0 on X. Then by the homogeneity of f 2 and φ, f 2 (x 0 ) = φ(x 0 ) and f 2 (x) φ(x) in a neighbourhood U of x 0. Thus φ is a local linear subgradient of f 2 at x 0. By Proposition 1 each local linear subgradient is also a global linear subgradient. Observe that its restriction to X gives a Φ-subgradient on X. Corollary 1. Let f be a periodic function with period 2π. If at each point t there is a local subgradient of f of the form a t sin t + b t cos t, where all a t, b t are bounded as functions of t, then at each t there is a global subgradient of this form. P r o o f. We simply rewrite Proposition 3 in polar coordinates. Corollary 2. Let f(t, s) be a function with period 2π with respect to t, π/2 s < π/2. If at each point (t, s) there is a local subgradient of f of the form a (t,s) sin s + b (t,s) cos s sin t + c (t,s) cos s cos t, where all a (t,s), b (t,s), c (t,s) are bounded as functions of (t, s), then at each (t, s) there is a global subgradient of this form. P r o o f. We simply rewrite Proposition 3 in spherical coordinates. Problem 1. Does the family Φ in Proposition 3 have the globalization property? Of course if an open set X R 2 is not connected Proposition 3 does not hold. Indeed, if X = X 1 X 2, where X 1, X 2 are disjoint and open in X, then the function { 1 for x X1, f(x) = 0 for x X 2

4 72 S. Rolewicz has local Φ-subgradient 0 at each point of X. But it is not a Φ-subgradient of f at x 0 for x 0 X 1. For connected sets the situation is more complicated. Of course in the case of the space R 1 each connected set is automatically convex and Proposition 3 holds. For R 2 we have the following Proposition 4. Let X be a simply connected non-convex open set in R 2. Let Φ be the restrictions of linear functionals to X. Then Φ does not have the globalization property. P r o o f. Since X is not convex and simply connected there is a half-plane H = {(x, y) : ax + by < 1} such that the intersection of H and X is not connected. We denote two components of X H by X 1 and X 2. Consider the intersection of the line L = {(x, y) : ax + by = 1} with X. We can find (x 1, y 1 ) X 1 L which is a boundary point of conv(x 1 L) and an interior point of conv(x L) on L. Let L 1 = {(x, y) : a 1 x + b 1 y = 1} be a line containing (x 1, y 1 ) and such that there is (x 2, y 2 ) X 1 with a 1 x 2 +b 1 y 2 > 1. The existence of such a line is easy to show. Let f(x, y) = max[0, a 1 x + b 1 y 1]. It is easy to see that f has a local Φ-subgradient at each point of X. On the other hand, f has no Φ-subgradient at any point of X such that a 1 x + b 1 y > 1, in particular at (x 2, y 2 ). As a consequence of Proposition 2 we obtain Proposition 5. Let Y be a convex subset of a linear topological space. Let Ψ be a class of linear functionals restricted to Y. Let X be a topological space and let h be a homeomorphism of X onto Y. Define Φ = {φ : φ(x) = ψ(h(x)), ψ Ψ}. Then Φ has the globalization property. P r o o f. Let f be a real-valued function on X. Suppose that φ is a local Φ-subgradient of f at x 0 X. Since h is a homeomorphism the image of an open set is open. Thus ψ(y) = φ(h 1 (y)) is a local Ψ-subgradient of f(h 1 (y)) at y 0 = h(x 0 ). Since this holds for all x 0 and h maps X onto Y, the function f(h 1 (y)) has a local Ψ-subgradient at each point of Y. Then it has a Ψ-subgradient, call it again ψ, at each y 0 Y, i.e. Thus f(h 1 (y)) f(h 1 (y 0 )) ψ(y) ψ(y 0 ) for all y Y. f(x) f(x 0 ) ψ(h(x)) ψ(h(x 0 )) for all x X and the function φ(x) = ψ(h(x)) Φ is a Φ-subgradient of f. Problem 2. Is it essential in Proposition 5 that the mapping h is oneto-one?

5 On a globalization property 73 Problem 3. Suppose that a family Φ of functions defined on a topological space X has the globalization property. Are there a convex set Y in a topological space and a homeomorphism h of X onto Y such that the functions φ(h 1 (y)) are linear? STEFAN ROLEWICZ INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES P.O. BOX WARSZAWA, POLAND Received on ; revised version on and

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