Convexity with respect to Beckenbach families
Mihály Bessenyei
joint work with Ágnes Konkoly and Bella Popovics
University of Debrecen, Institute of Mathematics
52nd International Symposium on Functional Equations
Bildungshaus Grillhof, Innsbruck (Austria), June 2229, 2014
National Excellence Program, Magyary Zoltán Postdoctoral Scholarship
Mihály Bessenyei
Convexity with respect to Beckenbach families
Preliminaries and motivations
Theorem (BaronMatkowskiNikodem, 1994)
I
f g I
Let be an interval, , : → R given functions. Then, there exists a convex
function : → R satisfying ≤ ≤ if and only if, for all , ∈ and
λ ∈ [0, 1],
λ + (1 − λ) ≤ λ ( ) + (1 − λ) ( ).
h I
f x
f h g
y
gx
xy I
gy
Theorem (Carathéodory, 1911)
H ⊂ Rn and p ∈ conv(H ), then there exist points p , . . . , pn of H such that
p ∈ conv{p , . . . , pn } holds.
If
0
0
Mihály Bessenyei
Convexity with respect to Beckenbach families
Preliminaries and motivations
Theorem (NikodemW¡sowicz, 1995)
I
f g I
Let be an interval, , : → R given functions. Then, there exists an ane
function ω : → R satisfying ≤ ω ≤ if and only if, for all , ∈ and
λ ∈ [0, 1],
λ + (1 − λ)
≤ λ ( ) + (1 − λ) ( ),
λ + (1 − λ)
≥ λ ( ) + (1 − λ) ( ).
I
f x
g x
f
g
y
y
Theorem (Helly, 1923)
xy I
gx
fx
gy
fy
n
If K is a collection of convex, compact subsets of Rn of which
each ( + 1)
T
member subcollection has a nonempty intersection, then K is also nonempty.
Mihály Bessenyei
Convexity with respect to Beckenbach families
Beckenbach families
Denition
A set F of real valued continuous functions dened on an interval I is called a
Beckenbach family if, for all points (x , y ) and (x , y ) of I × R with x 6= x ,
there exists exactly one member ϕ of F such that ϕ(x ) = y and ϕ(x ) = y .
A function f : I → R is called an F-convex function if for all x < x elements
of I and x ∈ [x , x ], we have the inequality f (x ) ≤ ϕ(x ) where ϕ ∈ F satises
ϕ(x ) = f (x ) and ϕ(x ) = f (x ).
1
1
2
2
1
1
1
2
1
1
1
1
2
2
2
2
2
2
Examples for Beckenbach families
x
x
F = {ϕ : R → R | ϕ( ) = α + β};
The linear hull of a Chebyshev-system;
The lines of the Moulton-plane.
Lemma
If a sequence of a Beckenbach family converges in two points, then it converges
uniformly on each compact subset of the domain.
Mihály Bessenyei
Convexity with respect to Beckenbach families
F-convex
sets
Denition
I
p x y
p x y
I
generalized segment
p
p
p p
x x
x x
[p , p ] := {(x , y ) | x = x = x , min{y , y } ≤ y ≤ max{y , y }},
[p , p ] := {(x , y ) | min{x , x } ≤ x ≤ max{x , x }, y = ϕp0 p1 (x )}.
Assume that the set [p , . . . , pn ] has already
S been dened and let pn ∈ I × R
be a given point. Then, [p , . . . , pn ] := {[p , pn ] | p ∈ [p , . . . , pn ]}. For a
subset H of I × R, the F-convex hull conv (H ) is given by
[
conv (H ) :=
{[p , . . . , pn ] | pk ∈ H , k = 0, . . . , n}.
Let F be a Beckenbach-family over the real interval . Let 0 = ( 0 , 0 ) and
× R. Under the
spanned
1 = ( 1 , 1 ) be given points of
by 0 and 1 we mean the set [ 0 , 1 ] given by (distinguishing the cases
0 =
1 and
0 6=
1 ):
0
0
1
0
1
1
0
0
1
1
0
0
1
,
1
+1
0
+1
0
+1
0
F
F
0
n∈N
K I
K
We say that ⊂ × R is F
is, if = convF ( ) holds.
K
-convex, if it coincides with its F-convex hull, that
Mihály Bessenyei
Convexity with respect to Beckenbach families
F-convex
sets
Theorem
I
Let F be a Beckenbach family over a real interval . Then,
(i) the F-convex hull of a nite set is invariant under permutations;
(ii)
(iii)
K is F-convex if and only if [p , p ] ⊂ K for all p , p ∈ K ;
the mapping conv : P(I × R) → P(I × R) is monotone and idempotent;
1
2
1
2
F
(iv) the intersection of F-convex sets is F-convex;
(v) the union of nested F-convex sets is F-convex;
(vi) for all
H ⊂ I × R,
\
conv (H ) =
{K ⊂ I × R | H ⊂ K , conv (K ) = K }.
F
F
Mihály Bessenyei
Convexity with respect to Beckenbach families
F-convex
Lemma
sets
X
X
Assume that Φ : P( ) → P( ) is a monotone, idempotent mapping of which
singletons are xed points. Then, for all ⊂ ,
\
Φ( ) = { ⊂ | ⊂ , Φ( ) = }.
H
H X
K X H K K
K
Proof
K X H KK
K
K
H K
H
K K
H
h H
{h} = Φ({h}) ⊂ Φ(H ).
Hence
H ⊂ Φ(H ). Therefore, due to the idempotency, Φ(H ) ∈ K, so we get
T
K ⊂ Φ(H ). The last statement of the previous theorem is an immediate
consequence of this lemma with Φ := conv , and using property (iii ).
Let K :=
⊂ | ⊂ , = Φ( ) . If ∈ K, then ⊂ holds;
T hence,
by the monotonicity, Φ( ) ⊂ Φ( ) = follows. That is, Φ( ) ⊂ K. For
the converse inclusion, take ∈ . Then the xed point property guarantees
F
Mihály Bessenyei
Convexity with respect to Beckenbach families
Combinatorial geometry of
Theorem
F-convex
structures
I
p p p p
AB
If F is a Beckenbach family over and 0 , 1 , 2 , 3 are pairwise distinct points
of × R, then there exists a partition { , } of the set { 0 , 1 , 2 , 3 } such
that convF ( ) ∩ convF ( ) 6= ∅.
I
A
B
Theorem
I
p p p p
If F is a Beckenbach family over a real interval and K is a collection of
F-convex subsets of × R such that, for all members T
0,
1,
2 ∈ K we have
K 6= ∅.
0 ∩
1 ∩
2 6= ∅ and K contains a compact set, then
K K K
I
Theorem
K K K
I H I
p p p H
If F is a Beckenbach family over a real interval and ⊂ × R, then, for all
element of convF ( ) there exists a subset { 0 , 1 , 2 } of , such that
∈ convF { 0 , 1 , 2 }.
p
p
p p p
H
Mihály Bessenyei
Convexity with respect to Beckenbach families
F-convex
functions
Theorem
I
f I → R is a given
If F is a Beckenbach family over an open interval and :
function, then the following statements are equivalent:
(i)
f
is F-convex;
(ii) for all elements
f | x0 x1
x
0
≤
x
1
<
x
2
≤
x
,
]
≥ ϕ|[x0 ,x1 ] ,
[
1
(iv)
(v)
I
of , we have the inequalities
f | x1 x2 ≤ ϕ| x1 x2 , f | x2 x3 ≥ ϕ| x2 x3
where ϕ ∈ F is dened by ϕ(x ) = f (x ) and ϕ(x ) = f (x );
for all element p ∈ I there exists a Beckenbach line ϕ such that ϕ ≤ f
and ϕ(p ) = f (p );
for all element p ∈ I , we have the representation
f (p) = sup{ϕ(p) | ϕ ∈ F, ϕ ≤ f };
epi(f ) is an F-convex set.
[
(iii)
3
,
]
[
1
,
]
[
2
,
]
[
,
]
2
In particular, F-convex functions are continuous at interior points of the
domain.
Mihály Bessenyei
Convexity with respect to Beckenbach families
Applications
Theorem
I
K I
g
If F is a Beckenbach family on an interval and ⊂ × R is a compact set,
then is F-convex if and only if there exists a compact subinterval and there
exist functions , : → R such that is F-convex, is F-concave, and the
representation = epi( ) ∩ hyp( ) holds.
K
f g J
K
f
g
f
Theorem
I
J
K
If F is a Beckenbach family over an interval and is an F-convex, compact
subset of × R, then
= convF (extrF ( )).
I
K
K
Theorem (NikodemPáles, 2007)
I
f g I → R are given
If F is a Beckenbach family over a real interval and , :
functions, then the following statements are equivalent:
h I → R such that f ≤ h ≤ g ;
for all elements x ≤ x ≤ x of I we have the inequality f (x ) ≤ ϕ(x ),
where ϕ ∈ F is determined by the properties ϕ(x ) = g (x ) and
ϕ(x ) = g (x ).
(i) there exists an F-convex function :
(ii)
0
1
2
1
0
2
0
2
Mihály Bessenyei
Convexity with respect to Beckenbach families
1
Bibliography
K. Baron, J. Matkowski, and K. Nikodem,
Math. Pannon. 5 (1994), no. 1, 139144.
A sandwich with convexity,
Generalized convex functions, Bull. Amer. Math. Soc.
E. F. Beckenbach,
43 (1937), 363371.
M. Bessenyei, Á. Konkoly, and B. Popovics, Convexity with respect to
Beckenbach families, manuscript.
J. Krzyszkowski, Generalized convex sets, Rocznyk Nauk.-Dydaktik Prace
Mat. 14 (1997), 5968.
Generalized convexity and separation
K. Nikodem, and Zs. Páles,
, J. Convex Anal. 14 (2007), no. 2, 239248.
theorems
A sandwich theorem and Hyers-Ulam
K. Nikodem, and Sz. W¡sowicz,
, Aequationes Math. 49 (1995), no. 1-2,
160164.
stability of ane functions
Mihály Bessenyei
Convexity with respect to Beckenbach families
Acknowledgement
Acknowledgement
This research was supported by the European Union and the State of Hungary,
co-nanced by the European Social Fund in the framework of
TÁMOP-4.2.4.A/2-11/1-2012-0001 `National Excellence Program'.
Mihály Bessenyei
Convexity with respect to Beckenbach families