Introduction: Notation, Elementary Results
We start this chapter by listing some basic concepts, which are or should be wellknown — but it is good sometimes to return to basics. This gives us the opportunity of making precise the system of notation used in this book. For example, some readers may
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We start this chapter by listing some basic concepts, which are or should be wellknown - but it is good sometimes to return to basics. This gives us the opportunity of making precise the system of notation used in this book . For example, some readers may have forgotten that "i.e," means id est, the literal translation of "that is (to say)". If we get closer to mathem atics, S\{ x} denotes the set obtained by depriving a set -1
S of a point xES. We also mention that, if f is a function, f (y) is the inverse image of y , i.e. the set of all points x such that f (x) = y. When f is invertible, this set is the singleton {j-I (y After these basic recalls, we prove some results on convex functions of one real variable. They are just as basic, but are easily established and will be of some use in this book.
n.
1 Some Facts About Lower and Upper Bounds 1.1 In the totally ordered set lR , inf E and sup E are respectively the greatest lower bound - the infimum - and least upper bound - the supremum - of a nonempty subset E , when they exist (as real numbers). Then, they mayor may not belong to E; when they do, a more accurate notation is min E and max E. Whenever the relevant infima exist, the following relations are clear enough: inf (E U F) = min {inf E , inf F}, } FeE ===} inf F ~ inf E , inf (E n F) ~ max {inf E , inf F} .
(1.1)
If E is characterized by a certain property P , we use the notation
E
= {r
E lR : r satisfies P} .
Defining (in lR considered as a real vector space) the standard operations on nonempty sets E + F := {r = e + f : e E E , f E F}, tE := {tr : r E E} for t E lR (the sign ":=" means "equals by definition "), it is also clear that
J. -B. Hiriart-Urruty et al., Fundamentals of Convex Analysis © Springer-Verlag Berlin Heidelberg 2001
2
O. Introduction: Notation, Elementary Results
inf (E + F) = inf E + inf F , } inf tE = t inf E if t > 0 , inf(-E) = -supE,
(1.2)
whenever the relevant extrema exist. The word positive means "> 0", and nonpositive therefore means "::::; 0"; same conventions with negative and nonnegative . The set of nonnegative numbers is denoted by JR+ and, generally speaking , a substar deprives a set of the point O. Thus, for example, N* ={1 ,2, . ..}
and
JR::
={tEJR : t>O}.
Squared brackets are used to denote the intervals of JR : for example ,
JR :J [c, b] = {t E JR : a
< t ::::; b} .
The symbol "L" means convergence from the right, the limit being excluded; thus, t -!- 0 means t ----+ 0 in IR; . The words "increasing" and" decreasing " are taken in a broad sense: a sequence (tk) is increasing when k > k' :::} tk ? tk' . We use the notation (tk), or (tkh, or (tkhE ]\/. for a sequence of elements tl, ia , . .. 1.2 Now, to denote a real-valued function
X 3x
f-7
f defined on a nonempty set X , we write f(x) E JR.
The sublevel-set of f at level r E JR is defined by
Sr(f) :={XEX: f(x) ::::;r} . If two functions
f and 9 from X to JR satisfy f(x) ::::; g(x)
for all x EX ,
we say that f minorizes 9 (on X), or that 9 majorizes Computing the number inf {f(x) : x E
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