Is every radiant function the sum of
quasiconvex functions?
Alberto Zaffaroni
Università di Lecce
Abstract
An open question in the study of quasiconvex function is the characterization of the class of functions which are sum of quasiconvex functions.
In this paper we restrict attention to quasiconvex radiant functions, i.e.
those whose level sets are radiant as well as convex and deal with the
claim that a function can be expressed as the sum of quasiconvex radiant
functions if and only if it is radiant. Our study is carried out in the
framework of Abstract Convex Analysis: the main tool is the description
of the supremal generator of the set of radiant functions, i.e. the class
of functions whose sup-envelope gives radiant functions, and of the relation between the elements in the supremal generators of radiant and
quasiconvex radiant functions.
Key Words: Radiant functions, Quasiconvex functions, Abstract Convexity
1
Introduction
The concept of a quasiconvex function has been introduced in the first half
of the last century as an important extension of convexity. Among the first
authors to investigate quasiconvexity, one finds the names of von Neumann,
de Finetti and Fenchel. Expecially in the last thirty years the research on this
topic has grown very extensively and the results obtained by a great number
of researchers are deep enough to elaborate what has been recently called
”Quasiconvex Analysis”. We refer to [6] and to the extensive bibliography
there included for an overview of results about quasiconvex functions.
Concerning the algebraic structure of the set of quasiconvex functions, it is
readily seen that this set forms a cone and an upper semilattice, in that quasiconvexity is preserved under positive scalar multiplication and under taking
suprema.
On the other hand, simple examples in IR show that the sum or the infimum
of quasiconvex functions need not be quasiconvex; consequently some effort
has been devoted in the related literature to characterize the functional class
containing sums of quasiconvex functions. Partial results in this direction
were obtained by Pearce and Rubinov [4], who characterize such class, limited
1
to functions of one real variable; see also Crouzeix and Lindberg [1], who
study the conditions under which the sum of quasiconvex functions is itself
quasiconvex.
Our attention is devoted to a special subset of quasiconvex functions, given
by those functions which have a global minimum at the origin. These functions
are characterized as the ones whose lower level sets are convex and include the
origin, i.e. they are convex and radiant. This class received great attention
mainly in connection with conjugation theory (see for instance [8, 5, 6] and
references therein) and Mathematical Economics.
If we consider functions of one variable belonging to this class, we obtain a
set which forms a closed convex conic lattice, i.e. it is also closed with respect
to sums and infima. The reason for this is that the family of intervals in IR
which contain the origin is closed with respect to both intersection and union
and the level sets of the sum and the infimum can be represented by means of
these operations.
√
Nevertheless simple examples in IR2 (consider f1 (x, y) = x, f2 (x, y) =
√
y, their sum and minimum) show that these properties fail in general in
greater dimensions.
In this paper we characterize the least convex conic lattice containing radiant quasiconvex functions as the set of functions - which we call radiant whose level sets are radiant, i.e. closed with respect to homotheties of modulus
smaller than one.
We study the class of radiant functions from the point of view of Abstract
Convexity (see [3, 9, 7]): the main tool to obtain our results is the description
of a supremal generator for the family of (l.s.c.) radiant functions, that is a
family C of ‘elementary’ functions such that, for every extended real valued
l.s.c. radiant function f , it holds
f (x) = sup{c(x) : c ≤ f, c ∈ C};
To obtain such description, we start from the study of the relevant separation
properties of radiant sets. In close analogy with closed convex sets, every
closed radiant set can be separated by points not belonging to it by means of
an open convex cone or, equivalently, by (positive) level sets of a superlinear
function, whereas open halfspaces and linear functionals are used for convex
sets.
The differences between support functions for quasiconvex and radiant
function follow the same analogy: they both take on just two values, the
greater of which corresponds to an open convex cone for a radiant function
2
and to an open halfspace for quasiconvex functions; the analysis of the relations
between the two is a key step to obtain our results.
An outline of the paper is the following: Section 2 is devoted to the definition and the study of some general properties of radiant functions; in Section
3 we recall some basic notion of Abstract Convex Analysis and apply them
to the description of radiant functions; the fundamental separation theorem
for radiant sets is given there in its geometric and analytic forms. Section 4
contains our main results about the relation between radiant functions and
sum of quasiconvex functions. Some conclusions are given in Section 5.
2
General properties of radiant functions
Here and in what follows, except when explicitly stated otherwise, X is a real
normed vector space, Bδ denotes the ball around the origin of radius δ and X 0
the topological dual of X, that is the normed vector space of linear continuous
functionals defined on X. A set K ⊂ X is a cone if x ∈ K implies αx ∈ K for
all α > 0.
We are interested in studying the properties of functions whose level sets
are radiant sets according to the following definitions.
Definition 2.1 A set A of a vector space X is said to be radiant if x ∈ A
and α ∈ [0, 1] imply αx ∈ A.
Definition 2.2 A function f : C ⊆ X → IR ∪ {±∞}, where C is a cone, is
said radiant if its lower level sets Sk (f ) = {x : f (x) ≤ k} are radiant for
every k ∈ IR.
It is easy to see that a convex set A ⊂ X is radiant if and only if it contains
the origin and consequently a quasiconvex function f defined on X is radiant
if and only if the origin belongs to every lower level sets, that is 0 is a global
minimum of f .
Let R(X) denote the class of all extended valued function with radiant
level sets defined on a space X. As it is easily checked, the following characterizations hold.
Proposition 2.3 A function f : C ⊆ X → IR ∪ {±∞}, where C is a cone, is
radiant if and only if, for every x ∈ C, the function fx : IR+ → IR given by:
fx (α) ≡ f (αx),
is non decreasing.
3
α≥0
Proof: Suppose that 0 ≤ α < β implies f (αx) ≤ f (βx) and take x ∈ Sk (f )
for some k. Then, for β = 1, we have the first part of the thesis. On the
other hand consider α < β and fix x ∈ X. If f (βx) = k, then βx ∈ Sk (f ) and
αx = (α/β)βx ∈ Sk (f ), which means that f (αx) ≤ k.
2
Following Proposition 2.3, with some abuse of terminology, we will sometimes say that a function f is increasing along rays to mean that it is radiant.
Proposition 2.4 A function f : C ⊆ X → IR ∪ {±∞}, where C is a cone, is
radiant if and only if its strict lower level sets Tk (f ) ≡ {x ∈ X : f (x) < k}
are radiant.
Proof: Let fx be nondecreasing for x ∈ Tk (f ) and take α < 1. Then f (αx) ≤
f (x) < k and Tk (f ) is radiant.
Let, on the other hand, Tk (f ) be radiant for every k and take x ∈ X and
α < β. If it holds f (αx) > f (βx), take k such that f (αx) > k > f (βx); in this
case we would have βx ∈ Tk (f ) and (α/β)βx = αx ∈
/ Tk (f ) which is absurd.
2
If the function f is radiant, then either f (0) = −∞ or 0 is a global minimum
point for f or f (x) = ∞ everywhere.
The usefulness of radiant functions in Mathematical Economics (particularly in Utility Theory) is shown in [12]
The set of radiant functions is stable under a number of operations; we
list some of them in the sequel, emphasizing the related algebraic properties
of the set R(X). Each of them is followed by the simple proof of the related
statement.
• R(X) is a cone, that is f ∈ R(X) implies αf ∈ R(X) for each α > 0.
Indeed Sk (αf ) = Sk/α (f ) for α > 0.
• R(X) is convex, i.e. fi ∈ R(X) for i = 1, 2 implies f1 + f2 ∈ R(X).
Indeed if f = f1 + f2 , then
Sk (f ) =
[
Sc (f1 ) ∩ Sk−c (f2 ),
c∈IR
with Sc (fi ) radiant for any c ∈ IR and i = 1, 2 and the family of radiant
sets is closed with respect to arbitrary intersection and union.
4
• R(X) is pointwise closed, i.e. it is closed with respect to pointwise
convergence. To see this, let f be the pointwise limit of the sequence
{fn } ⊆ R(X), i.e. f (x) = lim fn (x). Consider x ∈ Tk (f ) and α < 1. To
prove that f (αx) < k, observe that there exists k 0 such that f (x) < k 0 <
k and N such that fn (x) < k 0 for n > N . This yields
fn (αx) ≤ fn (x) ≤ k 0
and f (αx) = lim fn (αx) ≤ k 0 < k.
• R(X) is a complete lattice, that is the supremum and the infimum of
any family of radiant functions are radiant. Let {fu , u ∈ U } ⊆ R(X)
be any family of radiant functions and let f (x) = supu∈U fu (x) and
g(x) = inf u∈U fu (x). To see that f ∈ R(X) take x ∈ Sk (f ) and α < 1.
It follows
fu (αx) ≤ fu (x) ≤ sup fu (x) = f (x) ≤ k
u
and f (αx) = sup fu (αx) ≤ k.
To show that g ∈ R(X), consider x ∈ Tk (g) and α < 1. Since g(x) < k
there exists ū ∈ U such that the following holds:
k > fū (x) ≥ fū (αx) ≥ inf fu (αx) = g(αx).
u
• R(X) is closed with respect to the level sum, i.e. the operation which
associates to a pair of functions f, g ∈ R(X) the function whose level
sets are the algebraic sum of the level sets of the functions f and g. This
operation is defined as
+
F (x) = (f ∨ g)(x) = inf [f (x − w) ∨ g(w)],
w
where [r ∨ s] = max{r, s} for r, s ∈ IR and satisfies x ∈ Tk (F ) if and only
if x ∈ Tk (f ) + Tk (g), the latter being radiant as the sum of radiant sets.
• Performance function: if f : W × X → IR ∪ {±∞} is radiant, then
the performance (or marginal) function p(w) = inf x f (w, x) is radiant.
Indeed if p(w) < k then there exists x̄ ∈ X such that f (w, x̄) < k
and this implies f (γw, γ x̄) < k for all γ ∈ [0, 1] and hence p(γw) =
inf x f (γw, x) ≤ f (γw, γ x̄) < k.
5
3
Abstract Convexity of radiant functions
We have seen in Section 2 that the set R(X) is a closed convex conic complete
lattice containing the set Q(X) of quasiconvex functions with minimum point
at the origin. Our aim is to show that R(X) is the minimal set which such
properties. To do this we rely on the tools of Abstract Convex Analysis (we
mainly refer to [7], but see also [3] and [9]).
Given a set H of functions defined on the space X, a function f : X →
IR ∪ {±∞} is said to be Abstract Convex with respect to H (or H-convex for
short) if it holds
f (x) = sup{h(x)| h ≤ f, h ∈ H}.
(1)
This definition stems from the well-known characterization of a lower semicontinuous convex (sublinear) functions as the supremum of its affine (linear)
minorants and suggests that convex functions can be considered as emerging
from the interplay of linearity and sup-envelope representation. Decoupling
the two contributions, (1) can be used in order to characterize various classes
of nonconvex functions by describing, for each, its supremal generator H.
If one notice how closely the notions of subgradient and conjugate functions
are related to (1), it is not surprising that the approach of Abstract Convexity
proved to be such a useful way to extend many global properties of convex
functions to various classes of nonconvex functions (see [7] for a number of
examples in this direction).
For instance two subclasses of the family of quasiconvex radiant functions
can be described and analyzed by means of the tools of Abstract Convex
Analysis.
To generate the set Q(X) of all lower semicontinuous functions in Q(X)
one can use as a set of elementary functions the family S of linear two-steps
functions defined as follows [7]: for any linear continuous functional ` ∈ X 0
and any pair of numbers c, c0 ∈ IR ∪{−∞} with c ≥ c0 , let s`,c,c0 ∈ S be defined
as
(
c
`(x) > 1
s`,c,c0 (x) =
0
c
otherwise
We will say that S is a supremal generator for the class of l.s.c. quasiconvex radiant functions and will say that s ∈ S is a support function for
f if s ≤ f .
By considering the weak inequality `(x) ≥ 1 in the definition above, we
obtain a different class of elementary functions, which generate the family
of evenly quasiconvex radiant functions, that is those functions whose lower
6
level sets are evenly convex and radiant, evenly convex sets being those which
can be represented as intersections of open halfspaces. For instance all l.s.c.
quasiconvex radiant functions, but also all u.s.c. quasiconvex radiant functions belong to this class, though not every quasiconvex function is evenly
quasiconvex.
The first step to take if we want to study radiant functions in the framework
of Abstract Convexity is to describe the family of elementary functions which
generate them by means of sup-envelopes.
As for quasiconvex functions we can consider different families of elementary functions according to the various continuity properties we intend to
describe.
The family R≥ contains all functions rz,c,c0 of this type: for fixed z ∈ X
and c, c0 ∈ IR ∪ {−∞} with c ≥ c0 let
(
rz,c,c0 (x) =
c
c0
x = αz, α ≥ 1
otherwise
The family R> contains functions rz,c,c0 such that:
(
rz,c,c0 (x) =
c
c0
x = αz, α > 1
otherwise
Thus the difference lies in that rz,c,c0 (z) = c if r ∈ R≥ and rz,c,c0 (z) = c0 if
r ∈ R> .
The following results describes radiant functions.
Proposition 3.1 Let f : X → IR ∪ {±∞}. Then f is radiant if and only if
it is R≥ -convex.
Proof: It is trivial to see that any function in R≥ is radiant and, as we have
seen in Section 2, the supremum of radiant functions is still radiant.
Conversely for any function f ∈ R(X) and any x ∈ X, we can consider
the function rz,c,c0 ∈ R≥ with z = x, c = f (x) and c0 = f (0). Then we have
an elementary function which minorizes f and coincides with it at the point
x.
2
On the other hand the family R> generates the set of all increasing along
rays functions which are lower semicontinuous on every ray.
Proposition 3.2 Let f : X → IR ∪ {±∞}. Then f is radiant and lower
semicontinuous on every ray from the origin if and only if it is R> -convex.
7
Proof: Notice to start that all functions in R> are increasing along rays and
also radially lower semicontinuous and that both properties are mantained
by taking suprema. Thus every function which is sup-envelope of elements
of R> is radiant and radially l.s.c. To show the opposite relation, take any
radiant function f , which is also radially l.s.c. and any x ∈ X. It must
hold f (x) = sup{rz,c,c0 (x) : r ∈ R> , rz,c,c0 ≤ f }. Thus we need to show
that, for every k < f (x) there is a support function such that rz,c,c0 (x) ≥ k.
If f (x) = f (0) this is trivial by taking c = c0 = f (0) and any z ∈ X. If
f (x) > f (0), since f is radially l.s.c., there is a small segment I(x) on the ray
Rx = {y ∈ X : y = αx, α > 0}, containing x and excluding its endpoints,
with f (w) > k for all w ∈ I. Now choose z ∈ I such that z = βx with β < 1.
Thus, for c = k and c0 = f (0), we have rz,c,c0 ≤ f and rz,c,c0 (x) = k.
2
In both cases, if we limit the families R≥ and R> to include only elementary functions for which c, c0 ∈ IR, by taking suprema we obtain functions
which are bounded below.
To characterize in this framework lower semicontinuous radiant functions,
one should consider a family of l.s.c. functions; we will consider the following
family C of ‘conical’ two-step functions: let K be an open, convex cone in X,
z ∈ K, c ≥ c0 ∈ IR ∪ {−∞} and define
(
fz,K,c,c0 (y) =
c
c0
if y ∈ z + K
otherwise
To show that C is indeed a supremal generator for the set R(X) of l.s.c.
radiant functions defined on X, we will need some preliminary results of independent interest.
Definition 3.3 For a set A ⊆ X we call shadow of A the set
shw (A) = [1, +∞)A = {x ∈ X : x = λa, λ ≥ 1, a ∈ A}.
Proposition 3.4 For any closed and radiant set A ⊆ X and x ∈
/ A, there
exists an open convex cone K with x ∈ K and some β ∈ (0, 1) such that
A ∩ (βx + K) = ∅.
Proof: Since A is closed there exists some open ball around x, U (x), with
A ∩ U = ∅. Moreover, since A is radiant, then A ∩ shw U = ∅. Let K = cone U .
Then K is an open convex cone containing x. Moreover there exists β ∈ (0, 1)
8
such that βx ∈ U and βx + K ⊆ shw U . Indeed if y ∈ U and k ∈ K = cone U ,
then we can write k = αu with α > 0 and u ∈ U and
µ
y+k = y+αu = (1+α)
¶
1
α
y+
u = β(γy+(1−γ)u) = βu0 ∈ shw U
1+α
1+α
since β = 1 + α > 1 and γ = 1/1 + α ∈ (0, 1), so that u0 ∈ U , which is convex.
2
Proposition 3.5 The function f : X → IR is l.s.c. and radiant if and only if
it is C-convex.
Proof: It is trivial to see that each function in C is l.s.c. and radiant and that
the same holds for the supremum of any family of support functions.
To prove that every l.s.c. radiant function is indeed the supremum of its
support set in C, take f ∈ R(X) and fix x ∈ X. For every k < f (x) we must
exhibit an elementary function minorizing f and whose value at x is at least
k.
This is trivial if f (x) = f (0). Otherwise if f (x) > k > f (0), we can take
0
c = f (0) and c = k and observe that the set Sk (f ) is closed and radiant, with
x∈
/ Sk (f ); then apply Proposition 3.4 to obtain an open convex cone K and
z = βx. It holds z ∈ K and x ∈ z + K. Hence we have fz,K,c,c0 ≤ f and
fz,K,c,c0 (x) = k as we wanted.
2
Proposition 3.4 and the description of the elementary functions in C are
given in a geometric form based on the use of the open convex cone K. In
both cases we can give an alternative, more analytic form, based on superlinear functions. This is again in close analogy to the possibility of giving the
main separation theorems for convex sets in a geometric form (which refers to
halfspaces and hyperplanes) and in an analytic form (with linear functionals).
Proposition 3.6 If K 6= X is an open convex cone and z ∈ K, then there
exists a superlinear continuous function p : X → IR such that z + K = {x ∈
X : p(x) > 1}.
Proof: Since K is contained in an open halfspace and z ∈ K then 0 ∈ z − K
and there exists some positive δ such that the ball Bδ around 0 satisfies Bδ ⊆
(z − K), with (z − K) ∩ (z + K) = ∅.
Since the z + K is open and convex, the set L ⊆ X 0 of linear continuous
functionals which separate z + K from z − K is nonempty. Moreover, since
the set z + K excludes a ball around the origin, we can suppose that for every
9
` ∈ L it holds `(x) > 1 for all x ∈ z + C and `(x) ≤ 1 for all x ∈ Bδ . For the
same reasons, for every point x in the boundary of z + K there exists some
` ∈ L such that `(x) = 1.
For every x ∈ X let p(x) = inf `∈L `(x). Then p is superlinear and u.s.c.
by construction and satisfies z + K = {x ∈ X : p(x) > 1}. To show that
p is indeed continuous, notice that the norm of a linear functional, given by
k`k = sup{|`(x)| : kxk ≤ 1} can also be given (see [2]) by the reciprocal of
the distance from the origin of its 1-hyperplane, i.e. k`k−1 = dH1 (`) (0) where
H1 (`) = {x ∈ X : `(x) = 1} and dA is the distance function from the set
A ; moreover it holds dH1 (`) (0) ≥ δ for each ` ∈ L. Thus the set L is norm
bounded by M = 1/δ. This implies that p is bounded on the unit ball; indeed
for every x ∈ B = {kxk ≤ 1} we have:
p(x) = inf `(x) ≥ inf −k`k = −M.
L
L
Hence its effective domain coincides with X and p is continuous. Moreover
the set L is convex and weak* closed and hence it holds L = ∂p(0), i.e. the
set L is precisely the superdifferential of p at the origin.
2
The separation result given in Proposition 3.4 reads now as follows.
Corollary 3.7 For any closed and radiant set A ⊆ X and any point x ∈
/ A,
there exists a superlinear continuous function p : X → IR such that p(a) ≤ 1
for every a ∈ A and p(x) > 1.
Corollary 3.7 forms the basis of a polarity relation between closed radiant
subsets of X and some convex subsets of the space of continuous superlinear functionals defined on X and of a conjugation theory for l.s.c. radiant
functions, which is analyzed in another paper [11].
Corollary 3.7 extends to infinite dimensional spaces an analogous result by
Shveidel [10], which makes use of particular superlinear functions.
Proposition 3.8 [10] Let A ⊆ IRn be closed and radiant and let x ∈
/ A. Then
there exist n linearly independent linear functionals `1 , ..., `n such that
p(a) = min `i (a) ≤ 1, ∀a ∈ A
i
and
p(x) = min `i (x) > 1.
i
By means of Proposition 3.6 we can give a different supremal generator
for R(X) using superlinear functions. We will then denote by P the family of
elementary function written as
(
sp,c,c0 (x) =
c
c0
10
if p(x) > 1
otherwise
for some continuous superlinear function p : X → IR and c ≥ c0 ∈ IR ∪ {−∞}.
Proposition 3.9 The set P is a supremal generator for the family of l.s.c.,
extended real valued, radiant functions defined on X.
Proof: Since the function p is superlinear and continuous, the set S1 (p) is
radiant and closed; since the lower level sets of a given function s ∈ P are
either empty or the whole X or given by S1 (p), then every s ∈ P is lower
semicontinuous and radiant and the same holds for the supremum of any family
in P. To show that every l.s.c. radiant function f is indeed the supremum
of its minorants in P, one can take any x ∈ X, any k < f (x) and obtain a
minorant c ∈ C, as in Proposition 3.5, given by c(x) = k for x ∈ z + K and
c(x) = f (0) for x ∈
/ z + K for some open convex cone K and z ∈ K.
By Proposition 3.6 we can associate to the set z + K a continuous superlinear function p : X → IR such that p(x) > 1 for each x ∈ z + K and the
result is proved.
2
The preceeding results have a particularly simple form in the finite-dimensional
case. It follows from Proposition 3.8 that, when X = IRn , the superlinear function appearing in the construction of the functions in P is indeed the minimum
of n linearly independent vectors of IRn .
4
Monotonicity along rays and quasiconvexity
We consider in this section the class Q(X) of quasiconvex radiant functions
defined on the space X, i.e. those having convex radiant sublevel sets. It holds
Q(X) ⊂ R(X) and moreover Q(IR) = R(IR), since a subset A ⊂ IR is radiant
if and only if it is an interval containing the origin and this holds if and only if
A is convex and radiant. The set Q(X) enjoys some of the algebraic properties
of R(X): it is a pointwise closed cone and a complete upper semilattice in that
it is closed with respect to pointwise supremum.
The set Q(X) fails to be convex in that the sum of two quasiconvex radiant
function
is not necessarily
quasiconvex (let e.g. f = f1 + f2 with f1 (x, y) =
p
p
|x| and f2 (x, y) = |y|).
In the following we will analyze in greater detail the relation between the
sets R(X) and Q(X) of l.s.c. functions in R(X) and Q(X) respectively; our
main tool will be the relation existing between the support functions of the
elements of the two families.
11
This allows to prove the following representation result, showing that any
lower semicontinuous radiant function can be constructed by means of the
operations of sum and supremum starting from l.s.c. quasiconvex functions.
Theorem 4.1 Any given l.s.c. radiant function f : X → IR ∪ {±∞} is the
supremum of a family of l.s.c. functions, each of which is the sum of l.s.c.
quasiconvex radiant functions.
Proof: Given f ∈ R(X), let f¯ be defined as f¯(x) = f (x) for x 6= 0 and f¯(0) =
−∞. It holds f¯ ∈ R(X). Its elementary minorant sp,c,c0 ∈ P necessarily have
c0 = −∞. For each of them consider the superdifferential ∂p(0) = {` ∈ X 0 :
`(x) ≥ p(x)} and to each elementary function sp,c,c0 associate the following
family of linear two-step functions, indexed by ` ∈ X 0 : for ` ∈ ∂p(0) then
(
s` (x) =
0
−∞
if `(x) > 1
otherwise
if ` ∈
/ ∂p(0), then set s` (x) = 0, ∀x. In this way, we obtain a family s` , ` ∈ X 0
such that
X
sp,c,−∞ (x) = c · 1I(x) +
s` (x),
`
where 1I(x) is the function taking the value 1 for every x. To show that this is
indeed the case, notice the following equivalences:
X
s` (x) = 0
⇐⇒
s` (x) = 0, ∀` ∈ X 0
⇐⇒
`(x) > 1, ∀` ∈ ∂p(0)
⇐⇒
p(x) > 1,
`
where the sum is well defined, since the summands only take the values 0 and
−∞ and the last coimplication holds because the superdifferential of p at the
point x is nonempty and given by
∂p(x) = {` ∈ ∂p(0) : `(x) = p(x)};
thus, if it were p(x) ≤ 1 we could find some ` ∈ ∂p(0) such that `(x) ≤ 1.
In this way we have constructed the l.s.c. radiant functions sp,c,−∞ ∈ P and
it holds f¯(x) = sup{sp,c,−∞ (x) : sp,c,−∞ ≤ f¯} and moreover f = sup{f¯, f (0)}.
2
12
The opposite relation is not necessarily true, in that an arbitrary sum of
l.s.c. functions needs not be l.s.c. Such further result can be obtained when
X = IRn to show that R(X) is the smallest convex complete upper semilattice
containing Q(X). This relies on the possibility to represent l.s.c. radiant
function by means of special elementary functions, as shown in Proposition
3.8, which in turn allows to use only sums of a finite number of elements.
Theorem 4.2 R(IRn ) is the least convex complete upper semilattice containing Q(IRn ).
Proof: Notice first that any finite sum of l.s.c. quasiconvex radiant functions
is l.s.c. and radiant and this is still true by taking suprema. Conversely if
f ∈ R(IRn ) we can define the function f¯, as in the proof of Theorem 4.1,
as f¯(x) = f (x) for any x 6= 0 and f¯(0) = −∞ and, by Proposition 3.8,
f¯ is the supremum of its minorant of the type sp,c,c0 where c0 = −∞ and
p(x) = mini `(x), where `1 , ..., `n is a collection of linearly independent linear
functionals. It is easily seen that this function is the sum of exactly n linear
two-step function which, for i = 1, ..., n, are written as
(
s`i (x) =
c/n
−∞
if `i (x) > 1
otherwise
which are l.s.c. quasiconvex and radiant. Then f¯ is given by the sup envelope
of its support functions in P and f = sup{f¯, f (0)}.
2
With an analogous reasoning it is possible to show that any l.s.c. radiant
function can be constructed only by means of infima and suprema, starting
from a family of l.s.c. quasiconvex radiant functions.
Theorem 4.3 Any given l.s.c. radiant function f : X → IR ∪ {±∞} is the
supremum of a family of l.s.c. functions, each of which is the infimum of a
family of l.s.c. quasiconvex radiant functions.
Proof: It is known, by Proposition 3.9, that any function f ∈ R(X) is the
supremum of its minorant in P. Moreover any element sp,c,c0 ∈ P can be seen
as
sp,c,c0 (x) = inf {s` (x)},
`
where s` ∈ S is given by
(
s` (x) =
c
c0
if `(x) > 1
otherwise
13
if ` ∈ ∂p(0) and is given by s` (x) = c for every x if ` ∈
/ ∂p(0). Thus sp,c,c0 is
the infimum of a family of l.s.c. quasiconvex radiant functions.
2
The inverse of this result may fail in that the infimum of an arbitrary
family of l.s.c. function needs not be lower semicontinuous. On the other
hand the situation becomes neater when X = IRn since, as it is easy to see, the
mimimum of a finite number of function can be used instead of the infimum.
Theorem 4.4 R(IRn ) is the least set containing infima of a finite number of
elements in Q(IRn ) and arbitrary suprema.
Proof: It is enough to notice that any function cp,c,c0 ∈ P in the support set
of a function f ∈ R(IRn ) is given by
cp,c,c0 = min s`i ,c,c0 ,
i
where p(x) = mini `i (x).
5
2
Conclusions
The original aim of this study is the claim that any radiant function is the
sum of a (possibly infinite) family of quasiconvex radiant functions and that
the sum contains at most n elements if we consider a function defined on IRn ,
i.e. the set of radiant functions is the convex hull of the set of quasiconvex
radiant functions.
Although we have not been able yet to prove the claim in full generality,
we have shown in this paper some partial results in this direction, by refering
to lower semicontinuous functions.
Looking for some evidence that the claim be false, we tried to consider
weather the level sets of functions which are sum of quasiconvex radiant functions would enjoy some properties which make them in some way ’better behaved’ than those of functions in R(X). For instance the level sets Sk (f ) of
a quasiconvex function are convex and hence regular, i.e. if their interior is
nonempty then it holds Sk (f ) ⊆ cl int Sk (f ). To show that this property is
lost for the sum of quasiconvex radiant
function, one canp
consider the function
p
f = f1 + f2 , with f1 (x1 , x2 ) = |x1 | and f2 (x1 , x2 ) = |x2 | if |x2 | ≤ 1 and
f2 (x1 , x2 ) = 1 otherwise and consider the level set S1 (f ).
The same example shows that the level sets need not be radiative in the
sense of [7].
So the claim remains open for future research.
14
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