proceedings of the american mathematical society Volume 109, Number 4, August 1990 OPEN RELATION THEOREM WITHOUT CLOSEDNESS ASSUMPTION SZYMON DOLECKI (Communicated by William J. Davis) Abstract. It is shown that a convex relation such that the preimage of a bounded set has nonempty interior is lower semicontinuous throughout the al- gebraic interior of its domain. If, besides, the relation is closed-valued, then it is closed throughout the algebraic interior of its domain. 1. Introduction A well-known theorem of convex analysis says that if y is a topological vector space, /: Y —*R is convex and there exists an open set Q0 such that (1.1) SUp/ < +00, then / is continuous on the algebraic interior of its domain domf = {y: f(y) < +00} (e.g. [6]). We prove open relation and closed-graph theorems that are straightforward generalizations to convex relations of the above fact concerning convex functions. In particular, these theorems do not require graph-closedness assumptions. Let us recall that the open relation theorem has been established (as an ex- tension of the Banach open mapping [2]) by Ursescu [9] and, independently, by Robinson [7] under the provision of closed convex graph and refined by Borwein [3] who replaced closedness by convergent series closedness. (See also other developments in [4] by Borwein.) Those open relation theorems specialize in another classical fact: if / is a lower semicontinuous convex function, Y is barreled and int(dom f) / 0, then / is continuous throughout the algebraic interior of its domain (e.g. [6]). It is well known that this result may be reduced to that evoked at the very beginning by showing that int(dom /) ^ 0 implies (1.1) provided that Y is a Baire space. The relationship of our open relation theorem to the existing ones is analo- gous. Proofs of older theorems hinge on the fact that, for closed convex rela- tions, surjectivity when conjugated with the Baire property implies a bounded- Received by the editors April 10, 1989 and, in revised form, September 20, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46A55; Secondary 46A30. © 1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 1019 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1020 SZYMON DOLECK.I ness property. Our open relation theorem consists in the implication of open- ness by boundedness. But what seems to be entirely novel is that boundedness actually entails (graph-) closedness. We divide the two mentioned facts into portions that will appear as conse- quences of the results of §3. Recall that y belongs to the algebraic interior intaA of A if, for every k € Y , there exists tQ > 0 such that y + tk € A as 0 < t < t0 . 1.1. Corollary. If f: Y —*R is convex and there exists an open set Q0 such that (1.1) holds, then, for every y € intQ(dom/), there is a neighborhood Q of y such that supe / < +00. 1.2. Corollary. Iff: Y —►R is convex and there exist a neighborhood D of the origin and c0> 0 such that sup +D f < f(y0)+c0, then there is a neighborhood Q of y0, a neighborhood D of the origin and c > 0 such that, for each y € Q and every 0 < t < I, (1.2) sup f < f(y) + tc. y+tD 1.3. Corollary. If there exists an open set Q0 such that (1.1) holds, then f is upper semicontinuous throughout intQ(dom/). 1.4. Corollary. If there exists an open set Q0 such that (1.1) holds, then f is lower semicontinuous throughout intQ(dom/). 1.5. Corollary. // / is lower semicontinuous and Y is barreled, then f is continuous on int(dom/). 2. Link between functions and relations The link with the general theorems that will follow is through the epigraph of /, i.e. epif = {y, r) € Y x R : f(y) < r}. Of course, epif is a relation between Y and R, its inverse relation being the lower level relation (epi/)~r = {y.f(y)<r}. Let Y, Z be topological spaces. A relation Q c Y x Z is called lower semicontinuous at y0 if, for each open subset Q of z such that Qy0 f)Q ^0, there exists a neighborhood P of y0 for which P c Q~ Q. By definition, Q~ is open at y0 whenever Q is lower semicontinuous at y0 . It is said to be closed at y0 if, for every z ^ Qy0, there exist neighborhoods P of y0 and Q of z such that Q n ÍIP = 0 . Recall that (2.A) A function / is upper semicontinuous at yQ if and only if epif is lower semicontinuous at y0 . (2.B) A function / is lower semicontinuous at y0 if and only if epi / is closed at y0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OPEN RELATION THEOREM WITHOUT CLOSEDNESS ASSUMPTION 1021 Now let 7 be a topological vector space (2.C) A function / fulfills (1.2) if and only if there exist a ball B in R and a neighborhood D of the origin in Y such that y + tDc(epifT(f(y) + tB). (2.D) A function / fulfills (1.1) if and only if there exists a bounded subset H0 of R such that int(epi f)~H0 / 0. Finally, recall that (2.E) dom/=(epi/)"R. 3. Open and closed relation theorems Let X, Y be topological vector spaces and let A be a convex subset of XxY. 3.1. Proposition. Suppose that there is a bounded subset HQ of X such that int A//0 / 0. Then, for every y € intQ AX, there exists a bounded set H such that y eint AH. Proof. Let y0 € Y and D be a neighborhood of the origin of Y such that y0 + D c AH. Since y € intQ AX, there exists yx e AX and 0 < a < 1 such that y = ( 1 - a)y0 + ayx . Let xx € A~yx and H = ( 1 - a)HQ + axx . Then, by the convexity of A, y + (l-a)D = (l -a)(y0+D)+ayx C (1 - a)AHQ + aAxx C A[(l -a)H0 + axx]=AH, what completes the proof because H is bounded. □ 3.2. Theorem [3]. Let yQ € Ax0 and let V and W be balanced neighborhoods of 0 in Y and X, respectively, for which (3.1) yQ+WcA(x0 + V), then, for each e > 0 and balanced neighborhoods B and D of 0 in X and Y, respectively, such that V + eV c B and D + eD c W, the formula (3.2) y + tDcA(x + tB) holds for every (x, y) € (xQ+ sV) x (y0 + eD) n A and 0 < t < 1. Proof. Choose e > 0. Let D and B satisfy the assumptions. Then, for x €x0 + eV and y € y0 + eD, we have (in view of (3.1)) (3.3) y + Dcy0 + eD + Dcy0 + W cA(x0+V)cA(x + eV+V)cA(x + B). Let 0 < t < 1 . Since y € Ax , by (3.1) and by the convexity of A, we have y + tD = (1 -t)y + t(y + D) C (1 - t)Ax + tA(x + B) c A[( 1 - t)x + t(x + B)] = A(x + tB). a License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1022 SZYMON DOLECKI 3.3. Theorem. If there exists a bounded subset H0 of X such that intA//0 / 0, then A~ is lower semicontinuous on int a AX. Proof. Let y0 G intQ AX. By virtue of Proposition 3.1, there exists a bounded set H and a balanced neighborhood W of 0 such that y0 + W c A//. Let xQ € A~y0 and V be an arbitrary balanced neighborhood of the origin of X. Then there exists X > 1 such that xQ+ XV D H, that is, y0 + W c A(jc0+ AF). In view of Theorem 3.2 (applied in the special case of e = 0), we get that y0 + (1/X)W c A(*0 + F), that is A~ is lower semicontinuous at (y0, x0). G 3.4. Theorem. IfA~y0 is closed and there is a bounded subset H0 of X such that y0 € int AH0, then A~ is closed at y0. Proof. Let x <£ A~y0 . Then we may find a balanced neighborhood B of the origin in X such that (3.4) (x + 3B) n A"y0 = 0. By the convexity of A, for each k € Y and every 0 < t < 1 , (3.5) tA~ (y0 + k)+A-(yQ-tk)c(l+ t)A~ yQ. Let D be a balanced neighborhood of the origin in Y such that y0 + D c AHQ, that is, for each k € D, A~(y0 + k) n HQ ^ 0. Since HQ is bounded, there exists such a Z, 0 < r < 1, that tH0 c 5 and besides Dr G ß , so that (3.6) x + 2B c(l+t)(x + 3B). As a consequence of the former inclusion, 0 € tA~(yQ + k) + B and, in view of (3.5), for each k e D, (3.7) A-(y0-//c)c(l + 0A~y0 + 5. By (3.4) and (3.6), (1 + t)A~yQ+ B is disjoint from x + B and thus, by (3.7), A~(y0 - tk) n (x + B) = 0 for each k € D, proving the closedness of A~ at V n The boundedness assumption used in Theorems 3.2, 3.3, and 3.4 is a con- sequence of surjectivity in the case when one can apply a Baire argument.