On the quadratic support of strongly convex functions

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Int. J. Nonlinea Anal. Appl. 7 2016 No. 1, 15-20 ISSN: 2008-6822 electonic http://dx.doi.og/10.22075/ijnaa.2015.273 On the quadatic uppot of tongly convex function S. Abbazadeh a,b,, M.

1 Int. J. Nonlinea Anal. Appl No. 1, ISSN: electonic On the quadatic uppot of tongly convex function S. Abbazadeh a,b,, M. Ehaghi Godji a a Depatment of Mathematic, Faculty of Mathematic, Statitic and Compute Science, Semnan Univeity, Semnan , Ian b Young Reeache and Elite Club, Malaye Banch, Azad Univeity, Malaye, Ian Communicated by Themitocle M. Raia Abtact In thi pape, we fit intoduce the notion of c-affine function fo c > 0. Then we deal with ome popetie of tongly convex function in eal inne poduct pace by uing a quadatic uppot function at each point which i c-affine. Moeove, a Hye-Ulam tability eult fo tongly convex function i hown. Keywod: tongly convex function; Hahn Banach theoem; c-affine function; quadatic uppot MSC: Pimay 26A25; Seconday 39B Intoduction and peliminaie Duing the 20th centuy mathematician intoduced and invetigated many genealization of convexity. A it i well known, the notion of the claical convexity can be expeed in tem of affine function. It i well known that any convex function defined on a eal inteval admit an affine uppot at evey inteio point of a domain the convee i alo tue. In thi pape, we genealize thi idea to tong convexity. We fit intoduce the notion of c-affine function and then chaacteize the tongly convex function defined on eal inne poduct pace. In the following theoem, the convex veion of Hahn Banach theoem i given. Coeponding autho addee:.abbazadeh.math@gmail.com S. Abbazadeh, madjid.ehaghi@gmail.com M. Ehaghi Godji Received: Decembe 2013 Revied: Novembe 2014

2 16 Abbazadeh, Godji Theoem 1.1. Robet and Vebeg [11] Let f be convex on an open et U of a nomed linea pace L and let V 0 be a nontivial ubpace uch that V 0 U Ø. If A 0 : V 0 R i affine and A 0 x fx on V 0 U, then thee i an affine extenion A : L R of A 0 uch that Ax fx on U. In what follow, X,. i a eal inne poduct pace, D tand fo an open convex ubet of X and c i a poitive contant. A function f : D R i called tongly convex with modulu c if f tx + 1 ty tfx + 1 tfy ct1 t x y 2 fo all x, y D and t 0, 1. Stongly convex function wee intoduced by Polyak [9]. They have ueful popetie in optimization theoy. Fo intance, if f i tongly convex, then it i bounded fom below, it level et x I : fx λ ae bounded fo each λ and f ha a unique minimum on evey cloed ubinteval of I [11], p Nikodem et al. have obtained ome inteeting popetie of tongly convex function ee [3, 6, 7, 8]. Any tongly convex function defined on a eal inteval admit a quadatic uppot at evey inteio point of it domain. Lemma 1.2. Nikodem and Pále [7] A function f : D R i tongly convex with modulu c if and only if the function g : D R defined by g = f c 2 i convex. Lemma 1.3. Robet and Vebeg [11] Call f : a, b R tongly convex with modulu c if and only if fo each x 0 a, b, thee i a linea function T uch that fx fx 0 +T x x 0 +cx x 0 2. We ay a function f : D R ha uppot at x 0 D if thee i an affine function A : D R uch that Ax 0 = fx 0 and Ax fx fo evey x D. A convex function f : D R i chaacteized by having a line of uppot at each point of domain. Theoem 1.4. Robet and Vebeg [11] The function f : D R i convex if and only if f ha uppot at each point of D. We define the cla of c-affine function a follow. Definition 1.5. A function q : D R i called c-affine if q tx + 1 ty = tqx + 1 tqy ct1 t x y 2 fo all x, y D. Remak 1.6. A function f : D R ha a quadatic uppot at x 0 D if thee exit a c-affine function q : D R with qx 0 = fx 0 uch that qx fx fo all x D.

3 On the quadatic uppot of tongly convex function No. 1, The quadatic uppot of tongly convex function In ode to pove the fundamental theoem on the quadatic uppot of tongly convex function, we need a veion of the Hahn Banach theoem which we now pove. Theoem 2.1. Let Y be a ubpace of a finite dimenional vecto pace X and f be a tongly convex function with modulu c on Y. If g : Y [0, i c-affine and gx fx on Y, then thee i a c-affine extenion h : X [0, of g uch that hx fx on X. Poof. Define fx to be + if x X \ D and note that f now i tongly convex on all of X. Chooe a fixed w X \ V 0. Then fo x, y V 0, > 0 and > 0 + q 0x + + x + + y Multiplying by + give o + q c x y 2 0y = q f + x + + y = f fx w + + x w + + c x y 2 fy + w q 0 x + q 0 y fx w + fy + w gx, = q 0x fx w fy + w q 0y y + w + = hy,. It follow that up g inf h on V 0 P whee P denote the poitive eal line. Moeove, if x 0 V 0 D and 0 i mall enough o that both x 0 0 w and x w ae in D, then both gx 0, 0 and hx 0, 0 ae finite. Hence, up g and inf h ae finite. In paticula q 0 x fx w α fx + w q 0x fo x V 0, > 0 and > 0. Subtituting t = when t < 0 and t = fo t > 0 lead immediately to q 0 x + αt fx + tw fo all x V 0 and t R. Now, let V 1 = { x + tw : x V 0, t R }. V 1 i a ubpace which popely contain V 0. Define q 1 : V 1 R by q 1 x + tw = q 0 x + tα. Then q 1 i c-affine on V 1, q 1 = q 0 on V 0 and we have jut etablihed that q 1 x + tw fx + tw. If V 1 = X, ou theoem i poved. If not, we may poceed by mathematical induction to extend q 0 to all of X ince dim V n = dim V n A an application of Hahn Banach theoem, we how that a c-affine function defined on a ubpace of eal inne poduct pace X can be extended to a c-affine function on X with peeving of it oiginal nom.

4 18 Abbazadeh, Godji Coollay 2.2. Let Y be a ubpace of a finite dimenional eal inne poduct pace X and g : Y [0, be a g -affine function g = up { gx : x Y }. Then g can be extended to a g -affine function h : X R uch that h = g. Poof. Define f : X [0, by fx = g 1 + x + x 2 gx fx and f tx + 1 ty fo each x X. Note that = g 1 + tx + 1 ty + tx + 1 ty 2 t + 1 t + t x + 1 t y + t x t y 2 t1 t x y 2 g = t g 1 + x + x 2 = tfx + 1 tfy g t1 t x y t g 1 + y + y 2 g t1 t x y 2 which how that f i tongly convex with modulu g on X. By Theoem 2.1 thee exit a g -affine extenion h of g to all of X uch that hx fx hold fo all x X. Thi implie that h g. On the othe hand, g = up { gx : x Y } up { hx : x X } = h alo hold. So, g = h and h i the deied extenion of g to all of X. We ae now eady to pove the fundamental theoem on the quadatic uppot of a tongly convex function. Theoem 2.3. The function f i tongly convex with modulu c > 0 on an open convex et D of X if and only if f ha quadatic uppot at each point of D. Poof. Suppoe f ha uppot at each point of D. Let x 1, x 2 D and x 0 = tx tx 2 whee t [0, 1] and let q : D R be a c-affine function uch that uppot f at x 0. Then fx 0 = qx 0 = tqx tqx 2 ct1 t x 1 x 2 2 tfx tfx 2 ct1 t x 1 x 2 2 which etablihe the tong convexity of f. Now uppoe f i tongly convex on D and chooe x 0 D. Take u to be a unit vecto with the ame diection a x 0 o an abitay unit vecto if x 0 = 0, and define the vecto ubpace V 0 = {x 0 + tu : t R}. The open inteval I = {t : x 0 + tu D} contain 0 and we may define a convex function gt = fx 0 + tu in the now cutomay way. Accoding to Lemma 1.3 thee i a c-affine function q : R R uch that G0 = g0 and Gt gt on I. We ue thi G to define q 0 : V 0 R by q 0 x 0 + tu = Gt. Now q 0 i c quadatic affine, q 0 x 0 = fx 0 and q 0 x 0 + tu = Gt gt = fx 0 + tu fo x 0 +tu V 0 D. It follow fom Theoem 2.1 that q 0 can be extended to a function f : X R whee qx fx on D. That i, q i a quadatic uppot function fo f at x 0.

5 On the quadatic uppot of tongly convex function No. 1, Remak 2.4. The theoem above can be poved by uing Lemma 1.2 and 1.4. By Lemma 1.2 a function f : D R i tongly convex with modulu c > 0 if and only if the function g = f c. 2 i convex. By Lemma 1.4, g i convex if and only if g ha uppot at each point of D. Let A : D R be an affine function that uppot g at x 0. So Ax 0 = gx 0 and Ax gx = fx c x 2 fo evey x D. Then, function q : D R defined by qx = Ax + c x 2 i a quadatic uppot fo f at x 0. A an application, we obtain the following Hye Ulam type tability eult elated to tongly convex and c-affine function. Let ε > 0, c > 0, X, be a eal inne poduct pace and D be a convex ubet of X. We ay that a function f : D R i ε-tongly convex with modulu c if f tx + 1 ty tfx + 1 tfy ct1 t x y 2 + ε fo all x, y D and t 0, 1. The following lemma i ueful in ou invetigation. Lemma 2.5. Baon et al. [2] Real function f and g, defined on a eal inteval I, atify f 1 tx + 1 ty tg 1 x + 1 tg 1 y fo all x, y D and t [0, 1] if and only if thee exit a convex function h : I R uch that f h g. Lemma 2.6. Let f, g : D R be function with f g on D. Then g i c-affine if and only if thee exit a tongly convex function h : D R uch that f h g on D. Poof. The only if pat i obviou. To pove the if pat, aume that g i c-affine and conide the function f 1, g 1 : D R defined by f 1 x = fx c x 2, g 1 x = gx c x 2. Fom the c-affinene of g, we get f 1 tx + 1 ty = f tx + 1 ty c tx + 1 ty 2 g tx + 1 ty c tx + 1 ty 2 = tgx + 1 tgy ct1 t x y 2 c tx + 1 ty 2 = tg 1 x + 1 tg 1 y fo all x, y D and t 0, 1. Hence, by Lemma 2.5, thee exit a convex function h 1 : D R uch that f 1 h 1 g 1 on D. Define hx = h 1 x + c x 2, x D. Then, by Lemma 1.2, h i tongly convex with modulu c and f h g on D. Theoem 2.7. Let f : D R be ε-tongly convex with modulu c. Then thee exit a tongly convex function h : D R with modulu c uch that fo all x D. fx hx ε

6 20 Abbazadeh, Godji Poof. Let x 1, x 2 D. Define g : D R by gx = a + y, x + c x 2 fo a R and y D with gx 1 = fx 1 +ε and gx 2 = fx 2 +ε. One can eaily how that g i c-affine. Fo each z x 1, x 2, z = tx tx 2 and we have fz = ftx tx 2 tfx tfx 2 ct1 t x 1 x ε = tgx tgx 2 ct1 t x 1 x 2 2 = gz. Hence, accoding to Lemma 2.6 thee exit a function h : D R which i tongly convex with modulu c and uch that f h g on D. Since gx fx = ε, then a deied. Refeence fx hx ε [1] S. Banach, Su le fonctionelle linêale, Studia Math and [2] K. Baon, J. Matkowki and K. Nikodem, A andwich with convexity, Math. Pannonica [3] R. Ge and K. Nikodem, Stongly convex function of highe ode, Nonlinea Anal [4] H. Hahn, Übe lineae Gleichungyteme in lincae Räumen, J. Reina Angew. Math [5] E. Helly, Übe lineae Funktionalopeafionen, Sitzungbe. Math.-Natuwi. K1. Akad. Wi. Wion [6] J. Makó, K. Nikodem and Z. Pále, On tong α, F convexity, Math. Inequal. Appl [7] K. Nikodem and Z. Pále, Chaacteization of inne poduct pace by tongly convex function, Banach J. Math. Anal [8] K. Nikodem and Z. Pále, Genealized convexity and epaation theoem, J. Conv. Anal [9] B.T. Polyak, Exitence theoem and convegence of minimizing equence in extemum poblem with etiction, Soviet Math. Dokl [10] F. Riez, Sat cetain yteme d êquation fonctionelle et l appoximation de fonction continue, C. R. Acad. Sci Pai [11] A.W. Robet and D.E. Vabeg, Convex Function, Academic Pe, New Yok, 1973.

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