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Anal. Num´er. Th´eor. Approx., vol. 14 (1985) no. 1, pp. 1–6 ictp.acad.ro/jnaat

ON SOME GENERALIZATIONS OF CONVEX SETS AND CONVEX FUNCTIONS

ALEXANDRU ALEMAN (Cluj-Napoca)

1. INTRODUCTION

A set in a vector space is usually called convex if together with any two of its points it contains the whole interval joining them. At the same time, the applications of mathematics involve some extensions of this definition.

Namely, in connection with the introduction of locally convex topologies J. von Neumann [11] requires only the midpoint of the interval to belong to this set.

Then J.W Green and W. Gustin [6] and recently L.F. German and V.P. Soltan [5] claim that just the points dividing the interval in prescribed ratios remain in the set. I. Muntean [10] proves existence theorems of supporting hyperplanes to the sets which are convex in the latter sense. Finally, in establishing some fixed point theorems V.S. Shulman [13] introduces a concept of convexity stating that the intermediate points run a portion of a curve joining the end- points of the interval. We associate with these convexity notions for sets the corresponding convexity notions for real functions defined on such sets.

In this paper we study an attenuated convexity concept which includes the convexity notions introduced by J. von Neumann and J. W. Green and W.

Gustin, and we put out some relations between this concept and the concept of usual convexity. A set Y in a vector space over the filed Ror real numbers or the field Cof complex numbers is said to be:

convexif for each x, y∈Y and each pin ]0,1[ we have

(1.1) (1−p)x+py ∈Y;

p-convex, withp in ]0,1[,if for each x, y∈Y (1.1) holds;

weakly-convex if for each x, y ∈Y there exists a p in ]0,1[ such that (1.1) holds.

Every convex set isp-convex for eachpin ]0,1[,and everyp-convex set with a p in ]0,1[ is weakly-convex. As the following examples show, there exist weakly-convex sets which are neither convex norp-convex for anyp in ]0,1[.

Example 1.1. The set {0}∪]1,2] in R is weakly-convex without being p- convex for any pin ]0,1[.

Example 1.2. Every open set Y in a topological vector space is weakly- convex. Indeed, ifx, y∈Y,then the function p→(1−p)x+py, p∈[0,1],is continuous at the point p= 0 andY is a neighborhood ofx, hence there is a p in ]0,1[ such that (1−p)x+py∈Y.

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A real functionf defined on a setY in a vector space is said to be: convex if the set Y is convex and for each x, y∈Y and each pin ]0,1[ we have (1.2) f((1−px+py)≤(1−p)f(x) +pf(y) ;

p-convex, with p in ]0,1[ if the set Y is p-convex and for each x, y ∈Y (1.2) holds;

weakly-convex if for each x, y∈ Y there exists a p in ]0,1[ such that (1.1) and (1.2) hold.

In Section 2 of this paper we shall prove the convexity of weakly-convex closed sets, and the convexity of the closure of p-convex sets. Section 3 is devoted to the proof of an accessibility theorem for p-convex sets. In the last section we shall prove the convexity of the weakly-convex and lower- semicontinuous functions. By a counterexample we shall show that the the- orems of A. Ostrowski [12], M.R. Mehdi [8] and E.De´ak [3] concerning the convexity ofp-convex functions which are bounded on sets of positive measure or on sets of second with Baire property fail for weakly-convex functions.

2. THE CONVEXITY OF WEAKLY-CONVEX SETS

Theorem 2.1. Every weakly-convex closed set in a topological vector space X is convex.

Proof. Supposing the contrary, we can find a non-convex closed weakly- convex set Y in X. Then there are x0, y0 ∈ Y and p0 in ]0,1[ such that z(p0)∈/Y, where

z(p) = (1−p)x0+py0.

Since the functionp→z(p), p∈[0,1],is continuous atp=p0andZ =X\Y is a neighbourhood of z(p0),there exists aδ >0 such that

(2.1) z(p)∈Z for all pin ]p0−δ, p0+δ[.

Denotea0 = supA where

A={a∈[0,1] :z(p)∈Z for all pin [p0, a]}.

From (2.1) we easily derive that p0 < a0. We shall show thaty =z(a0)∈Y. Suppose the contrary, i.e., y /∈Y. Then a0 <1 and we can find δ0 >0 with a00<1 such that

(2.2) z(p)∈Z for all p in ]a0−δ0, a00[

Let a1 ∈ A with a0−δ0 < a1 ≤ a0. Then a2 = a0+ δ20 ∈ A since z(p) ∈Z when p ∈ [p0, a1], and z(p) ∈ Z when p ∈]a1, a2[⊂]a0−δ0, a00[ by (2.2).

Therefore, a2 ∈ A and we arrive at the contradiction a2 ≤ a0 < a2. Hence y∈Y.

Further denoteb0 = infB where

B ={b≥0 :z(p)∈Z for all pin [b, p0]}.

As before, we have b0< p0 and x∈Y where x=z(b0).

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Now, we can prove (1−q)x+qy /∈ Y for all q in ]0,1[, in contradiction with the hypothesis thatY is weakly-convex. To this end we first remark that (1−q)x+qy=z(pq),wherepq = (1−q)b0+qa0∈]b0, a0[. There are a∈A and b∈B such that b0 ≤b < pq < a ≤a0. If pq ≥p0,by the definition of A we have z(pq)∈Z, and ifpq< p0, by the definition of B we have z(pq)∈Z.

Hence (1−q)x+qy /∈Y, and the proof of Theorem 2.1 is achieved.

Remark 2.2. a) When X is the Euclidean space Rn with finite dimension n,Theorem 2.1 has been established by V.F. Dem’janov and L.V. Vasil’ev [4], p. 16.

b) Since the above proof uses only the continuity of the function p → (1−p)x0 +py0p ∈ [0,1], with fixed x0 and y, Theorem 2.1 remains valid for topological vector groups of special type (cf. [9]).

c) As Example 1.1 shows, Theorem 2.1 fails for weakly-convex sets which

are not closed.

It is well known that the closure of every convex set in a topological vector space is convex (cf.[2], p.57). The following corollary shows that this result preserves forp-convex sets too.

Corollary2.3. The closure of everyp-convex setY in a topological vector spaces in convex.

Proof. The adherence ¯Y is p-convex since

(1−p) ¯Y +pY¯ = (1−p)Y +pY¯ ⊂(1−p)Y +pY ⊂Y .¯ By Theorem 2.1, ¯Y is convex.

When p= 12 this corollary has been proved by J. von Neumann [11].

3. THE CONVEXITY OFp-CONVEX SETS.

As Example 1.2 shows, Theorem 2.1 is false for open sets. However, this theorem remains true for p-convex open sets. In proving this we are in need of the following lemma of J.W. Greem and W. Gustin [6]:

Lemma3.1. Letp∈]0,1[.Denote by(Pn)n≥1the sequence of sets inductively defined as follows: P1 = {0, p,1} : if Pn = {0, p(1)n , p(2)n , . . . , p(2nn),1} where 0< p(1)n < . . . < p(2nn)<1,has been already defined, put

Pn+1=Pn∪ {(1−p)p(k−1)n +pp(k)n : 1≤k≤2n+ 1},

where p(0)n = 0 and p(2nn+1) = 1. Then the setP =∪{Pn:n≥1} is dense in the interval [0,1].

Lemma3.2. If Y is a p-convex set in a vector space, thenY isq-convex for each q in the setP in Lemma 3.1.

Proof. If suffices to prove thatq ∈Pn, n≥1,implies

(3.1) (1−q)Y +qY ⊂Y.

When n= 1 and q∈P1 ={0, p,1},the inclusion (3.1) is immediate.

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Suppose (3.1) is valid for an integer n≥ 1. Let q ∈ Pn+1. We can admit that q /∈Pn so thatq has the form

q = (1−p)p(k−1)n +pp(k)n for a k∈ {1, ..,2n+ 1}.

If z ∈ (1−q)Y +qY, hence z = (1−q)x+qy with x, y,∈ Y, then z = (1−p)u+pv, where

u=

1−p(k−1)n

x+p(k−1)n y∈

1−p(k−1)n

Y +p(k−1)n Y ⊂Y and

v=

1−p(k)n

x+p(k)n y ∈Y

sincep(k−1)n , p(k)n ∈Pn. Thereforez= (1−p)u+pv∈(1−p)Y +pY ⊂Y.

We now state an accessibility result for p-convex sets.

Theorem 3.3. If Y is a p-convex set in a topological vector space X, x ∈ intY, y∈Y¯ and0< α <1,then

(3.2) (1−α)x+αy∈intY.

Proof. First we prove that ifx0intY, y0 ∈Y and 0< α <1, then

(3.3) (1−α)x0+αy0∈Y.

By Lemma 3.2, there is a positive numbersinP such thatα <1−s.

Since the function t→ x0+t·αs (y0−x0), t∈ [0,1], is continuous at t= 0 and Y is a neighbourhood of x0, there exists aγ ≤ αs, s < γ≤1, such that (3.4) x0+t·αsgg·s y0−x0

∈Y for all 0,γ·xα

.

There is at0 in ]0,γ·sα [ such that

(3.5) (1→δt0)x0t0y0 ∈Y where δt0 = (1−t0)1−sα . Indeed, the interval I =

h

1−γ·sα α

1−s,1−sα i

⊂ [0,1] has a positive length hence by Lemma 3.1, there existsq ∈P ∩int I. It follows that one can find a T0∈]0,γ·sα [ withq=δt0 ∈P and, by Lemma 3.2, we obtain (1−δt0)x0t0y0 ∈ Y.

From (3.4) and (3.5) we deduce (1−r)x0+ry0 ∈Y, where r= αs ·t0, and (1−α)x0+xy0 = (1−s) [(1−δt0)x0t0y0] +s

(1−r)x0+ry0

∈Y because Y is s-convex. Thus (3.3) is proved.

Now, we are in a position to prove (3.2). Denotez= (1−α)x+xy. Since the functionf :X→X given by

f(u) =u+ 1−α1 (z−u)

is continuous at u = y and f(y) = x, there exists a neighbourhood V of y such that f(V) ⊂ int Y. There is a y0 ∈ Y ∩V because y ∈ Y¯. From f(y0) ∈ f(V) ⊂int Y and (3.3) we derive z = (1−α)f(y0) +αy0 ∈ int Y. This proves the theorem.

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It is well known that the interior of a convex set in a topological vector space is convex (cf. [2], p.55). The following corollary shows that this result

holds even forp-convex sets.

Corollary3.4. The interior of everyp-convex in a topological vector space X is convex.

When X =C Corollary 3.4 has been established by D.A. Horowitz, D.A.

Rose and E.B. Saff [7].

4. THE CONVEXITY OF WEAKLY-CONVEX FUNCTIONS

The epigraph of a real functionf defined on a set Y is the set Ef ={(x, z)∈Y ×R:f(x)≤z}.

We need the following well-known lemmas (cf. [1], pp.75-76):

Lemma 4.1. A real function defined on a convex set in a vector space is convex if and only if its epigraph is convex.

Lemma 4.2. If a real function f is lower-semicontinuous on a topological space X, then its epigraph is closed in the topological product X×R.

Theorem 4.3. Let f be a real function defined on a closed set Y in a topological vector space. Iff is weakly convex and lower-semicontinuous, then f is convex.

Proof. First we prove that the epigraphEf is weakly-convex. Let (x1, z1)∈ Ef and (x2, z2)∈Ef. Since f is weakly-convex, there exists a p in ]0,1[ such that (1−p)x1+px2 ∈Y andf((1−p)x1+px2)≤(1−p)f(x1) +pf(x2)≤ (1−p)z1+pz2, hence ((1−p)x1+px2,(1−p)z1+pz2)∈Ef.

Now, by Lemma 4.2 and Theorem 2.1, Y and Ef are convex, hence, by

Lemma 4.1,f is convex.

Corollary 4.4. Let f be a real function defined on a closed convex set in a topological vector space. If f is wealky-convex and lower-semicontinuous, thenf is convex.

Remark 4.5. When the weakly-convexity is replaced by the stronger con- dition of p-convexity and the lower-semicontinuous is weakened in different ways, Corollary 4.4 remains still true. More precisely, the convexity of every real function f which is 12 - convex on an interval in R has been established by A. Ostrowski [12] whenf is bounded on a set of positive measure, and by M.R. Mehdi [8] when f is bounded on a set second category having the Baire property. An extension of last results top-convex functions has been given by E. De´ak [3].

However, as the following example shows, the results of A. Ostrowski and M.R. Mehdi fail when the considered functions are only weakly-convex.

Example4.6. The functionf :R→R, defined buyf(x) = 1 ifxis rational and f(x) = 0 if x is irrational, satisfies the conditions of A. Ostrowski and

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M.D. Mehdi and it is weakly-convex. We shall show that f is not p-convex for any p in ]0,1[. Supposing the contrary, there exists a pin ]0,1[ such that for every x, y ∈Rthe inequality (1.2) holds. If p is irrational, use (1.2) with x= 1−p1 and y= 1p to arrive at the contradiction

1 =f(2) =f((1−p)x+py)≤1−pf(x) +pf(y) = 0.

If p is rational, use (1.2) with x = √

2 and y = p−1p

2 to arrive at the contradiction.

1 =f(0) =f((1−p)x+py)≤(1−p)f(x) +pf(y) = 0.

REFERENCES

[1] Barbu, V. and Precupanu, T.,Convexity and Optimization in Banach Spaces Ed.

Academiei, Bucharest, 1978.

[2] Bourbaki, N., El´´ements de math´ematiques. Livre V: Espaces vectoriels topologiques, Hermann, Paris, 1966.

[3] eak, E., Uber konvexe und interne Funcktionen, sowie eine gemeinsame Verallge-¨ meinerung von beiden,Ann. Univ. Budapest. E¨otv¨os Sect. Math. 5, 109–154 (1962).

[4] Dem’janov, V.F. and Vasil’ev, L.V., Nondifferentiable Optimization (in Russian), Nauka, Moscow, 1981.

[5] German, L.F. and Soltan, V.P., Quasiconvexity in linear and metric spaces (in Russian), Investigations in functional analysis and differential equations, “Shtiintsa”

Kishinew, 1981, pp. 19–24.

[6] Green, J.W. and Gustin, W., Quasiconvex sets. Canadian J. Math., 2, 489–507 (1950).

[7] Horowitz, D.S. Rose, D.A.andSaff, E.B.,A comment on the convexity condition, Indian J. Math. 18, 159–164 (1976).

[8] Mehdi, M.R.,On convex functions,J. London Math. Soc.39, 321–326 (1964).

[9] Muntean, I., Sur la nontrivialit´e du dual des groupes vectoriels topologiques,Mathe- matica (Cluj)14, 259–262 (1972).

[10] Muntean, I., Support points of p.-convex sets. Proc. Coll. Approx. Optimiz., Cluj- Napoca, October 25-27, 1984, pp. 293–302.

[11] Neumann, J. von,On complete topological spaces,Trans. Amer. Math. Soc.,37, 1–20 (1935).

[12] Ostrowski, A.,Uber die Funcktionalgleichung der Exponentialfunktion und verwandte¨ Funktionalgleichungen, Jahresb. Deutsch. Math. Ver.38, 54–62 (1929).

[13] Shul’man, V.S.,Fixed points of linear-fractional transformations (in Russian), Funk- tsional. Anal. i Prilozhen.,14, 93–94 (1980).

Received 11.XII.1983

Universitatea Babe¸s-Bolyai Facultatea de matematic˘a 3400 Cluj-Napoca ROM ˆANIA

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