Convexity and unique minimum points

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1 Convexity and unique minimum points Josef Berger and Gregor Svindland February 17, 2018 Abstract We show constructively that every quasi-convex, uniformly continuous function f : C R with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory. 1 Introduction Let (X, d) be a compact metric space. The infimum of a uniformly continuous function f : X R is given by inf f = inf {f(x) x X}. An element x of X is a minimum point of f if f(x) = inf f. The function f has at most one minimum point if d(x, y) > 0 inf f < f(x) inf f < f(y) for all x, y X. In Bishop s constructive mathematics [6, 7, 8], the framework of this paper, the following statements are equivalent: (I) Brouwer s fan theorem for detachable bars. (II) Every positive-valued, uniformly continuous function on a compact metric space has positive infimum. 1

2 (III) Every uniformly continuous function on a compact metric space which has at most one minimum point has a minimum point. The equivalence of (I) and (II) was proved in [2, 10] and the equivalence of (I) and (III) was proved in [1, 2, 11]. In [3], we proved (II) for quasi-convex functions whose domain C is a convex compact subset of R m. Quasi-convex means that f(λx + (1 λ)y) max (f(x), f(y)) for all λ [0, 1] and x, y C. That result was crucial for the constructive treatment of the fundamental theorem of asset pricing in [4] and corresponds to a constructively valid version of the fan theorem, see [5]. In this paper which is a sequel of [3] we show (III) for quasi-convex functions whose domain C is a convex compact subset of R m (Theorem 1) and generalise this result to finite-dimensional normed spaces (Theorem 2). As applications we obtain a supporting hyperplane theorem (Proposition 1) and a result on strictly quasi-convex functions (Proposition 2). Moreover, we obtain a new short proof of the fundamental theorem of approximation theory (Proposition 3). 2 Unique minimum points In this section, we prove the following theorem. Theorem 1. Let C be a convex compact subset of R m. Then every quasiconvex, uniformly continuous function f : C R with at most one minimum point has a minimum point. To this end, let C be a convex compact subset of R m. Fix a quasi-convex, uniformly continuous function f : C R which has at most one minimum point. Without loss of generality, we may assume that inf f = 0. For a subset S of C let f S denote the restriction of f to S. Lemma 1. Fix convex compact subsets A, B of C such that d(a, b) > 0 for all a A and b B. Then inf(f A ) > 0 or inf(f B ) > 0. Proof. The set A B is a convex compact subset of R 2m, and the function F : A B R, (a, b) max (f(a), f(b)) is positive-valued, quasi-convex, and uniformly continuous. By [3, Theorem 1] we can conclude that ε := inf F > 0. Since inf(f A ) > 0 inf(f A ) < ε 2

3 and inf(f B ) > 0 inf(f B ) < ε, this implies that inf(f A ) > 0 or inf(f B ) > 0. For x R m and j {1,..., m} the j-th component of x is denoted by x j. Lemma 2. For each ε > 0 there exists C C such that (i) C convex and compact (ii) inf(f C ) = 0 (iii) x, y C j (x j y j < ε). Proof. Consider the first coordinate. Since the projection mapping x x 1 is uniformly continuous and uniformly continuous images of totally bounded sets are totally bounded [8, Proposition 2.2.6], the set D = {x 1 x C} is totally bounded. Set ι = inf D and η = sup D. If η ι < ε, set C = C. We may therefore assume that ι < η. Define s = ι + 1(η ι) and t = ι + 2 (η ι). 3 3 By [3, Proof of Lemma 4], the sets and A = {x C x 1 s} B = {x C x 1 t} are convex and compact. For a A and b B we have d(a, b) > 0. By Lemma 1, we can conclude that In the first case, set and in the second case set inf(f A ) > 0 or inf(f B ) > 0. (1) C = {x C x 1 s} C = {x C x 1 t}. The the set C fulfils the properties (i) and (ii). Iterating this procedure, also over the coordinates, we eventually obtain a set C which fulfills (iii) as well. The diameter of a compact set S is defined by diam S = sup {d(x, y) x, y S}. 3

4 of Theorem 1. By Lemma 2, we can construct a sequence (C n ) of compact subsets of C such that (a) n (C n+1 C n ) (b) lim n diam C n = 0 (c) n (inf(f C n ) = 0). For each n, fix x n C n with f(x n ) < 1/n. The sequence (x n ) is a Cauchy sequence and its limit is a minimum point of f. 3 Applications 3.1 Finite-dimensional normed spaces A normed space V is finite-dimensional if either V = {0} or else there exist b 1,..., b m V such that the linear mapping κ : R m V, λ m λ i b i i=1 is bijective. (Injective in the sense that λ > 0 implies κ(λ) > 0.) In this case, both κ and its inverse κ 1 are uniformly continuous. See [6, 7, 8] for more information on finite-dimensional normed spaces. In view of the definition of a finite-dimensional normed space, we obtain a straightforward generalisation of Theorem 1. Theorem 2. Let C be a convex compact subset of a finite-dimensional normed space. Then every quasi-convex, uniformly continuous function f : C R with at most one minimum point has a minimum point. 3.2 Supporting hyperplanes A subset C of a normed space X is strictly convex if λa + (1 λ)b C for all a, b C with d(a, b) > 0 and all λ ]0, 1[. The set C, the interior of C, is defined as usual: x C ε > 0 y X (d(y, x) < ε y C) The following lemma is easy to prove. 4

5 Lemma 3. Fix a subset C of X. (a) If C is convex and open, then it is strictly convex. (b) If C is strictly convex and closed, then it is convex. Proposition 1. Let C be a compact, strictly convex subset of a finite dimensional normed space V. Let g : V R be a linear function, and v an element of V with g(v) > 0. Then the restriction of g to C has a minimum point z. Proof. Let f denote the restriction of g to C. Note that linear functions are quasi-convex. Fix a, b with d(a, b) > 0. Set c = (a + b)/2. Since C is strictly convex, there exists δ > 0 such that c δv C. We obtain f(c δv) < f(c) max (f(a), f(b)). Thus f has at most one minimum point. By Theorem 2, f has a minimum point. In the situation of Proposition 1, the set {x V g(x) = g(z)} is called a supporting hyperplane of C. 3.3 Strictly quasi-convex functions Let C be a convex subset of a normed space. A function f : C R is strictly quasi-convex if f(λx + (1 λ)y) < max (f(x), f(y)) for all λ ]0, 1[ and x, y C such that x y > 0. Lemma 4. Every strictly quasi-convex function is quasi-convex. Proof. Fix a strictly quasi-convex function f : C R. As a first step, we show that x, y C λ ]0, 1[ (f(λx + (1 λ)y) max (f(x), f(y))). (2) To this end, fix x, y C and λ ]0, 1[ and assume that f(λx + (1 λ)y) > max (f(x), f(y)). (3) 5

6 If x y > 0, we have f(λx + (1 λ)y) < max (f(x), f(y)) by the strict quasi-convexity of f. This contradicts (3), thus we can conclude that x = y. This implies that (3) is false. Therefore, we have proved (2). As a second step, we show that f is quasi-convex. Fix x, y C and λ [0, 1]. We either have λ < 1 or else λ > 0. Without loss of generality, we may assume that λ < 1. Assume (3). If λ > 0, then (2) implies that f(λx + (1 λ)y) max (f(x), f(y)), which contradicts (3). Thus λ = 0. This implies that (3) is false, therefore we have shown f(λx + (1 λ)y) max (f(x), f(y)). Remark 1. In a constructive proof, case distinctions are only allowed when we can decide which one of the cases holds. However, this restriction no longer applies when we prove a negated statement. This follows from the scheme ( A A). (4) Having (4) in mind, we observe that the proof of Lemma 4 kan be rephrased as follows. Assume that f is strictly quasi-convex. Fix x, y C and λ [0, 1]. We have to show that f(λx + (1 λ)y) max (f(x), f(y)). (5) This is the negation of (3). If x y = 0 or λ {0, 1}, the statement (5) holds anyway. In the case λ ]0, 1[ and x y > 0, the statement (5) follows from the quasi-convexity of f. Since strictly quasi-convex functions have at most one minimum point, Theorem 2 and Lemma 4 imply the following result. Proposition 2. Let C be a convex compact subset of a a finite-dimensional normed space. Then every strictly quasi-convex, uniformly continuous function f : C R has a minimum point. 6

7 3.4 Approximation theory Let Y be a subset of a normed linear space X. Let g : Y R be a function. An element y of Y is a minimum point of g if x Y (g(y) g(x)). The function g has at most one minimum point if x, y Y (d(x, y) > 0 z Y (g(z) < max (g(x), g(y)))). These definitions are compatible with those given in Section 1. For a X let fa Y be the function : Y y d(y, a). f Y a The set Y is quasiproximinal if for every a X the implication f Y a has at most one minimum point f Y a has a minimum point is valid. As a consequence of Theorem 2, we obtain the the constructive fundamental theorem of approximation theory from [9]. Proposition 3. Every finite-dimensional subspace V of a normed space X is quasiproximinal. Proof. Since {0} is quasiproximinal, we may assume that the dimension of V is positive. Fix a X. Set f = fa V and suppose that f has at most one minimum point. Since V is a positive-dimensional linear space, there exists b V such that ε := f(b) = d(b, a) > 0. The triangle inequality yields Set v V (d(v, b) > 2 ε f(v) > f(b)). (6) V 0 = {v V d(v, b) 3 d(a, b)}. The set V 0 is compact, see [8, Corollary 4.1.7], and convex. The function g = f V 0 is uniformly continuous and quasi-convex. By (6), g has at most one minimum point. Theorem 2 implies that g has a minimum point v 0. Applying (6) once again, we can conclude that v 0 is a minimum point of f. 7

8 References [1] Josef Berger, Douglas Bridges, and Peter Schuster. The fan theorem and unique existence of maxima. Journal of Symbolic Logic, 71(2): , [2] Josef Berger and Hajime Ishihara. Brouwer s fan theorem and unique existence in constructive analysis. Mathematical Logic Quarterly, 51(4): , [3] Josef Berger and Gregor Svindland. Convexity and constructive infima. Arch. Math. Log., 55(7-8), [4] Josef Berger and Gregor Svindland. A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov s principle. Ann. Pure Appl. Logic, 167(11): , [5] Josef Berger and Gregor Svindland. Constructive convex programming. In Klaus Mainzer, Peter Schuster, and Helmut Schwichtenberg, editors, Proof Computation Digitalization in Mathematics, Computer Science and Philosophy. World Scientific Publishing Co. Pte. Ltd., Singapore, forthcoming. [6] Errett Bishop and Douglas Bridges. Constructive Analysis. Springer- Verlag Berlin Heidelberg, [7] Douglas Bridges and Fred Richman. Varieties of Constructive Mathematics. Cambridge Univ. Press, [8] Douglas Bridges and Luminita Vîţă. Techniques of Constructive Analysis. Springer, [9] Douglas S. Bridges. A constructive proximinality property of finitedimensional linear subspaces. Rocky Mountain Journal of Mathematics, 11(4): , [10] William Julian and Fred Richman. A uniformly continuous function on [0, 1] that is everywhere different from its infimum. Pacific J. Math., 111(2): , [11] Ulrich Kohlenbach. Effective Moduli from Ineffective Uniqueness Proofs. An Unwinding of de La Vallée Poussin s Proof for Chebycheff Approximation. Ann. Pure Appl. Logic, 64(1):27 94,