On approximately convex functions

PROCEEDINGSOF THE AMERICANMATHEMATICALSOCIETY Volume 118, Number 1, May 1993 ON APPROXIMATELY CONVEX FUNCTIONS C. T. NG AND K. NIKODEM (Communicated by William J. Davis) Abstract. The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex functions and to approximately Wright convex functions. Let X be a real vector space, D be a convex subset of X, and £ be a nonnegative constant. A function f:D ->R is said to be e-convexif f(tx + (1 - t)y) < tf{x) + (1 - t)f{y) + e for all x,y &D and f£[0, 1] (cf.[2]); e-Wright-convexif f{tx + (1 - t)y) + /((1 - t)x + ty) < f(x) + f{y) + 2e for all x,y €D and t e[0, 1]; e-midconvex if f{^)< \{f{x) + f{y)) + e for all x,y eD. Notice that e-convexity implies e-Wright-convexity, which in turn implies e-midconvexity, but not the converse. The usual notions of convexity, Wrightconvexity, and midconvexity correspond to the case e = 0. A comprehensive review on this subject can be found in [1, 6, 8-10]. The Bernstein-Doetsch theorem relates local boundedness, midconvexity, and convexity (cf. [6, 10]). In order to extend this result to approximately midconvex functions, we first specify the assumptions on the topology 3~ to be imposed on X: the map (t, x, y) -+ tx + y from lxlxl-»l is continuous in each of its three variables. Here the scalar field R is under the usual topology. In former literature the topology S~ is called semilinear (cf. [4, 5, 7]). These assumptions are weaker than those for X to be a topological vector space. The finest JonI is formed by taking all subsets A c X with the property that if xo € A , x e X, then there exists a 3 > 0 such that tx + (1 - t)xo e A for all t e ] - S, S[. In earlier literature such sets A are called algebraically open [11] (cf. also [3-5, 7])- Lemma 1. If D is convex and f:D —>R is e-midconvex, then (1) f(k2-"x + (1 - k2-")y) < k2~nf{x) + (1 - kl~n)f{y) + (2 - 2~n+x)e for all x,yeD, «eN = {l,2,...}, and k&{0, 1,..., 2"}. Received by the editors August 8, 1991; presented to the 96th Annual Meeting of the AMS on January 19, 1990. 1991 Mathematics Subject Classification. Primary 26B25, 26A51; Secondary 39B72. Key words and phrases. Approximately convex functions, Bernstein-Doetsch theorem. This research is supported by an NSERC of Canada grant. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 103 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 104 C. T. NG AND K. NIKODEM Proof. We proceed by induction. For n = 1, the inequality is clear. Assume that (1) holds for some n e N. Let x, y e D and k e {0, 1, ... , 2"+1} be arbitrarily given. By appropriately labelling x and y we may assume that k < 2" . Then we get /(*2—»* + (i - *2—V) = / ((*2-* + (i-*2-")J0 + r) <5/(fc2-"* + (l-fc2-")y) + i/(y) + e < i[fc2"V(^) + (1 - *2-«)/(y) + (2 - 2-"+')£]+ I/(y) + e = k2-n~lf{x) + (1 - fc2-"-1)/(j) + (2 - 2-")e as required. This proves the lemma. Lemma 2. Lef D be open and convex. If f: D —► R is e-midconvex and locally bounded from above at a point x0 6 D, then it is locally bounded from below at this point. Proof. Let U c D be an open set containing xo on which f(x) < M. Let V :— U n (2xo - U). Then V is an open set containing xo . Let x e V be given, and let x' = 2xo - x . Then x' e U, and /(*o) = / (fL^) Hence f(x) > 2/(xo) - /(x') bounded from below on V . < j/(*) + j/***) + «• - 2e > 2/(xn) - Af - 2e, proving that / is Lemma 3. Let D be open and convex. If f:D -+ R is e-midconvex and locally bounded from above at a point of D, then it is locally bounded from above at every point of D. Proof. Assume that / is bounded from above on an open set U c D containing Xo. Let x £ D be arbitrarily given. Since D is open, there exist a point z e D and a number n e N such that x = 2-"x0 + (1 - 2_")z. Put V := 2~nU + (1 - 2~")z . Then V is open and contains x . For every v e V , v = 2~"u + (1 - 2~n)z for some u e U. Hence, by Lemma 1, we get f(v) < 2~"f(u) + (1 -2~")/(z) + 2e . The boundedness of / from above on V now follows from that of / on U. This proves the local boundedness of / from above at x . Lemma 4. Let D be open and convex. If f:D -»R is e-midconvex and locally bounded from below at a point of D, then it is locally bounded from below at every point of D. Proof. Assume that / is bounded from below on an open U c D containing Xo, and let x 6 D be arbitrarily given. Since D is open, there exist a point z e D and a number n e N such that x0 = 2_"x + (1 - 2~")z. Let V := (2"U + (1 - 2")z) n D. Then V is an open neighbourhood of x . If v e V, then u := 2~"v + (1 - 2_B)z e U, and so by Lemma 1, f{u) < 2~nf(v)+ (1 - 2~n)f{z) + 2e . The boundedness of / from below on U now implies that of / on V. This proves the local boundedness of / from below at x . Lemma 5. Let D be a convex subset of X . If f: D —>R is e-midconvex and S-convex, then it is 2e-convex. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON APPROXIMATELYCONVEX FUNCTIONS 105 Proof. Let x ^ y in D be arbitrarily fixed. By assumption f(tx + (1 - t)y) < tf(x) + (1 - t)f{y) + <5 for all / E [0, 1]. First, for t e [0, j], we obtain /(fx + (1 - f)y) = f{\[2tx + (1 - 2/)y] + iy) <\f{2tx + {\-2t)y) + \f{y) + e < \[2tf{x) + (1 - 20/C) + S] + i/(y) + e = */(*)+ (l-f)/lv) + <Si, where <5i = 5/2 + e. By symmetry in x and y, the above extends to all t e [0, 1], yielding the fact that / is ^i-convex. Iterating this scheme, we get that / is £„-convex for n = 2, 3, ... , where dn = \8n-\ +e. Since Sn —>2e as n -» oo, we obtain the conclusion that / is 2e-convex. Theorem 1. Let D c X be open and convex. If f:D —► R is e-midconvex and locally bounded from above at a point of D, then f is 2e-convex. Proof. By Lemmas 2 and 3, / is locally bounded from both sides at every point in D. Let x ^ y be arbitrarily given in D. The segment [x, y] = {tx + (l-t)y:t e[0, 1]} is the image of the compact interval [0, 1] under the continuous map t —► tx + (1 - t)y, and so [x, y] is compact. The local boundedness of / at every point in D implies that / is bounded on [x, y], say by M. This implies that the restriction of /to [x, y] is 2M-convex. By Lemma 5, applied to /on [x, y], we get f(tx + (1 - t)y) < tf{x) + (1 - t)f{y) + 2e for all t e [0, 1]. As x, y are arbitrary, this proves that / is 2e-convex on D. Corollary 1. Let D be an open convex subset of R", and let f:D -►R be emidconvex. If f is bounded from above on a set A c D of positive Lebesgue measure, then it is 2e-convex. Proof. Assume that /(x) / (^y^) < M for all x e A . Then - \f{x) + \f{y) + E - M + e for a11x' y £ A Since, by the theorem of Steinhaus, \(A + A) has nonempty interior, the local boundedness of / from above follows. Theorem 1 now yields the conclusion. Remark. The assumption that D is open in Theorem 1 is not redundant. We give an example. Let D c R2 be the closed half plane {(x, y) e R2:y > 0}, and let /:D -►R be given by f{x, y) - 0 if y > 0, and /(x, y) = \a{x)\ if y = 0. Here u:R-»R is a discontinuous additive map. Then / is bounded locally at each point interior to D and is midconvex on D; however, / is not convex on the x-axis and is, therefore, not convex on D. Lemma 6. Let / c R be an interval. ///:/-» R is e-midconvex on I and 2e-convex in the interior of I, then f is 2e-convex on I. Proof. We may suppose that / is not degenerated, to be interesting. Let x / y be given in /. Consider z = tx+( 1-t)y for given t 6 ]0, 1[. Let u = (x+z)/2 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 106 C. T. NG AND K. NIKODEM and v = (z+y)/2. Then u, v are interior to /, and z = tu + (l -t)v . Hence, by 2e-convexity in the interior of /, we get /(*)< */(«) + (l-*)/(«)+ 2«. Since f(u) < [/(x) + /(z)]/2 + e and f(v) < [f(z) + f(y)]/2 + e by emidconvexity on /, we obtain /<*)<t [m +2 m+«]+a - o [M±m.+,]+2s. This simplifies to f(z)<tf(x) + (l-t)f(y) + 6e. As t € ]0, 1[ is arbitrary, this proves that / is 6e-convex on /. By Lemma 5, / is 2e-convex on /. Theorem 2. Let D c X be convex, and suppose that the boundary of D contains no proper segment [a, b] = {ta + (1 - t)b:t e [0, 1]} where a ^ b in D. If f:D -> R is e-midconvex and is locally bounded from above at a point interior to D, then f is 2e-convex. Proof. By Lemma 3, applied to the restriction of / to the interior of D, we get the local boundedness of / from above at every interior point of D. To show that / is 2e-convex, we need to show that for an arbitrary given proper segment [a, b] c D, /is 2e-convex on [a, b]. Consider pulling [a, b] back to [0, 1] via g:[0, l]->[a,b], g{t) = ta + (l_-t)b. Also consider J:= fog. Since [a, b] contains interior points of D, /is locally bounded from above at some interior point of [0, 1]. Applying Theorem 1 to /, we get that / is 2e-convex in ]0, 1[. Applying Lemma 6, we get the 2e-convexity of / on [0, 1]. This in turn yields that / is 2e-convex on [a, b]. Example. Let D be a closed ball in R" (with the usual topology). Then every e-midconvex function /: D —>R, locally bounded from above at a point interior to D, must be 2e-convex. This observation extends to closed balls in a strictly convex real normed linear space. Lemma 7. Let D c X be open and convex. If f:D —>R is e- Wright-convex and locally bounded from below at a point of D, then it is 2e-convex. Proof. Let x ^ y in D be arbitrarily fixed. We need to show that f{tx + (1 - t)y) < tf(x) + (1 - t)f(y) + 2e for all t e [0, 1]. This is an observation on the one-dimensional line passing x and y; we can formally pull the problem back to the real field as follows. Consider E = {t e R:tx + (1 - t)y e D} and g:E -►R given by g(t) = f(tx + (1 - t)y). Since D is open and convex, so is E c R. By Lemma 4, the local boundedness of / from below at one point extends to every point of D, leading to the local boundedness of g from below at every point of E. Since [0, 1] is compact, g is bounded from below on [0, 1]. The e-Wrightconvexity passes onto g. In particular, we have g(l - t) + g(t) < g(0) + g(\) + 2e forallie[0, 1]. As g(t) is bounded from below over all / e [0, 1], the above implies that g(l - t) is bounded from above over all t e [0, 1]. Thus g is bounded from License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON APPROXIMATELY CONVEXFUNCTIONS 107 above on [0, 1]. It follows from Theorem 1 that g is 2e-convex on E. Hence f(tx + (l-t)y) = g(t) = g(t-l + (l-t)-0) <tg(l) + (l- t)g(0) + 2e = tf(x) + (1 - 0/00 + 2e, as required. This proves the lemma. Theorem 3. Let D c X be convex. If f:D —>R is e-Wright-convex and locally bounded from below at an interior point of D, then it is 2e-convex. Proof. Suppose xo is an interior point of D and / is locally bounded from below at xo . From Lemma 7, it follows that / is 2e-convex in the interior of D. Let [x, y] with x ^ y be a given proper segment of D. We need to show that / is 2e-convex on [x, y]. There are two possibilities. First consider the case where [x, y] contains an interior point of D. Then, by Lemma 4, / is locally bounded from below at a point of [x, y]. Evidently, this implies it is locally bounded from below at a point in ]x, y[ := {tx + (1,- t)y: 0 < t < 1}. Applying Lemma 7 to / on ]x, y[, or to its pull back on ]0, 1[ if necessary, we obtain that / is 2e-convex on ]x, y[. Further, by Lemma 6, we obtain that / is 2e-convex on [x, y]. Second, consider the case where [x, y] is on the boundary of D. In this case consider the triangle with vertices xo, x , and y . By e-midconvexity of / we get / (^j1) < \f(xo) + \f(z) + e for all z e [x, y]; but the segment {^Xo + \z: z e [x, y]} is in the interior of D and is compact; thus / is bounded from below on this segment. The above inequality implies that / is bounded from below on [x, y]. Thus, applying Lemma 7, we first get that / is 2e-convex on ]x, y[ and, further by Lemma 6, obtain that / is 2e-convex on [x, y]. This completes the proof. Remarks. The above results remain valid when openness of a convex set D is replaced by its openness relative to the manifold it generates. Theorems 1 and 3 are most forceful when the topology is the topology of algebraically open sets. Theorem 1 reduces to a result obtained by Kominek [3, Theorem 2] when e = 0. The ratio 2e in these two theorems is the best possible, as the following example illustrates. Let /: R -> R, /(x) = 0 for x < 0, and /(x) = 1 for x > 0. Then / is e-midconvex with lowest 8=1/2. It is e-convex with lowest e = 1. In Theorem 1 boundedness from above cannot be replaced by boundedness from below. For example, f(x) = \a(x)\, where a:R —► R is a discontinuous additive map, is midconvex on R, and is locally bounded from below. Yet / is not convex. References 1. E. F. Beckenbach, Convexfunctions, Bull. Amer. Math. Soc. 54 (1948), 439-460. 2. D. H. Hyers and M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828. 3. Z. Kominek, On additive and convexfunctionals, Rad. Mat. 3 (1987), 267-279. 4. _, Convex functions in linear spaces, Prace Nauk. Uniw. Slask. Katowic. 1087 (1989). 5. Z. Kominek and M. Kuczma, Theorems of Bernstein-Doetsch. Piccard and Mehdi and semi- linear topology,Arch. Math. 52 (1989), 595-602. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 108 C. T. NG AND K. NIKODEM 6. M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's equation and Jensen's inequality, PWN and Uniwersytet Slaski, Warszawa-Krakow- Katowice, 1985. 7. _, An example of semilinear topologies, Stochastica 12 (1988), 198-205. 8. C. T. Ng, On midconvex functions with midconcave bounds, Proc. Amer. Math. Soc. 102 (1988), 538-540. 9. _, Functions generating Schur-convex sums, General Inequalities 5 (Proc. 5th Internat. Conf. on General Inequalities, Oberwolfach 1986), International Series of Numerical Mathematics, Vol. 80, Birkhauser Verlag, Basel and Boston, 1987, pp. 433-438. 10. A. W. Roberts and D. E. Varberg, Convex functions, Academic Press, New York and Lon- don, 1973. 11. F. A. Valentine, Convex sets, McGraw-Hill, New York, 1964. Faculty N2L 3G1 of Mathematics, Department of Mathematics, University Technical of Waterloo, University, Waterloo, Ontario, PL-43-309 Bielsko-Biala, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Canada Poland