DTIC ADA640823: A General Theory of Almost Convex Functions (Preprint)

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A General Theory of Almost Convex Functions
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14. ABSTRACT
Let ??m = {(t0, . . . , tm) ??? Rn+1 : ti ??? 0 m i=0 ti = 1} be the standard m-dimensional simplex. Let ???
= S ??? ??? m=1 ??m then a function h: C ??? R with domain a convex set in a real vector space is
S-almost convex iff for all (t0, . . . , tm) ??? S and x0, . . . , xm ??? C the inequality h(t0x0 + ?? ?? ?? +
tmxm) ??? 1 + t0h(x0) + ?? ?? ?? + tmh(xm) holds. A detailed study of the properties of S-almost convex
functions is made. It is also shown that if S contains at least one point that is not a vertex, then an extremal
S-almost convex function ES: ??n ??? R is constructed with the properties that it vanishes on the vertices
of ??m and if h: ??n ??? R is any bounded S-almost convex function with h(ek) ??? 0 on the vertices of ??n,
then h(x) ??? ES(x) for all x ??? ??n. In the special case S = {(1/(m+1), . . . , 1/(m+1))} the barycenter of
??m very explicit formulas are given for ES and ??S(n) = supx?????n ES(x). These are of interest as ES
and ??S(n) are extremal in various geometric and analytic inequalities and theorems.
15. SUBJECT TERMS
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
A GENERAL THEORY OF ALMOST CONVEX
FUNCTIONS.
S. J. DILWORTH, RALPH HOWARD, AND JAMES W. ROBERTS
(cid:1)
Abstract. Let ∆m ={(t0,...,tm)∈Rn+1 :ti ≥0, (cid:2)mi=0ti =1}
be the standard m-dimensional simplex. Let ∅ (cid:3)= S ⊂ ∞m=1∆m,
then a function h: C → R with domain a convex set in a real
vector space is S-almost convex iff for all (t0,...,tm) ∈ S and
x0,...,xm ∈C the inequality
h(t0x0+···+tmxm)≤1+t0h(x0)+···+tmh(xm)
holds. A detailed study of the properties of S-almost convex
functions is made. It is also shown that if S contains at least
one point that is not a vertex, then an extremal S-almost con-
vex function ES: ∆n → R is constructed with the properties
that it vanishes on the vertices of ∆m and if h: ∆n → R is any
bounded S-almost convex function with h(ek) ≤ 0 on the ver-
tices of ∆n, then h(x)≤ES(x) for all x∈∆n. In the special case
S ={(1/(m+1),...,1/(m+1))}thebarycenterof∆mveryexplicit
formulas are given for ES and κS(n)=supx∈∆nES(x). These are
of interest as ES and κS(n) are extremal in various geometric and
analytic inequalities and theorems.
Contents
1. Introduction. 2
1.1. Definition and basic properties. 3
1.2. A general construction for the extremal S almost convex
set on a simplex. 7
1.3. Bounds for S-almost convex functions and the sharp
constants in stability theorems of Hyers-Ulam type. 16
2. General results when S is compact. 19
2.1. Mean value and semi-continuity properties. 19
Date: January 31, 2001.
2000 Mathematics Subject Classification. Primary: 26B25 52A27; Secondary:
39B72 41A44 51M16 52A21 52A40.
Key words and phrases. Convex hulls, convex functions, approximately convex
functions, normed spaces, Hyers-Ulam Theorem.
The research of the second author was supported in part from ONR Grant
N00014-90-J-1343 and ARPA-DEPSCoR Grant DAA04-96-1-0326.
1
2 DILWORTH, HOWARD, AND ROBERTS
2.2. Simplifications in the construction of E∆n when S is
S
compact. 23
3. Explicit Calculation of E∆n and κ (n) when S the
S S
barycenter of ∆ . 27
m
3.0.1. The formula for E∆n. 29
S
3.0.2. Calculation of κ (n). 35
S
References 39
1. Introduction.
Let C be a convex set in a real vector space and let h: C → R. Then
according to Hyers and Ulam [5] for ε > 0, h is ε-approximately convex
iff
(1.1) h((1−t)x+ty) ≤ ε+(1−t)h(x)+th(y), for all t ∈ [0,1].
In [5] they show that if h is ε-approximately convex and C ⊆ Rn then
there is a convex function g: C → R and a constant C(n) only depend-
ing on the dimension so that |h(x) − g(x)| ≤ 1C(n)ε. In a previous
2
paper we show the sharp constant is C(n) = (cid:8)log n(cid:9)+ (cid:1)2(n+1−2(cid:1)log2n(cid:2)).
2 n+1
(Here (cid:8)·(cid:9) is the floor, or greatest integer function, and (cid:10)·(cid:11) is the ceiling
function.) Inthepresentpaperwegeneralizethenotionofapproximate
convexity and give the sharp constants in theorems the corresponding
Hyers-Ulamtypetheorem. Thisisdonebyfindingtheextremalapprox-
imately convex function on the simplex that vanishes on the vertices.
To put the these problems in a somewhat larger setting. First by re-
placing h by ε−1h in (1.1) there is no loss of generality in assuming that
ε = 1. Then many natural notions of generalized convexity(cid:1)are covered
in the following. Let ∆ = {(t ,...,t ) ∈ Rm+1 : t ≥ 0, m t = 1}
m 0 m i i=0 i
be the standard m-dimensional simplex.
1.1. Definition. Let V a vector space o(cid:2)ver the reals and let ∅ (cid:3)=
C ⊆ V be a convex set and let ∅ (cid:3)= S ⊆ ∞ ∆ . Then a function
m=1 m
h: C → R is S-almost convex on C iff for all (t ,...,t ) ∈ S and
0 m
x ,...,x ∈ C the inequality
0 m (cid:3) (cid:5)
(cid:4)m (cid:4)m
h t x ≤ 1+ t h(x )
i i i i
i=0 i=0
holds. We denote by
AlmCon (C) := {h : h is S-almost convex on C}
S
the set of almost convex functions h: C → R. (cid:1)
ALMOST CONVEX FUNCTIONS 3
The case of S = ∆ corresponds to the case studied by Hyers and
1
Ulam [5] and others (cf. the book [4] for more information and refer-
ences). When S = {(1/2,1/2)} the S-almost convex functions are just
the functions that satisfy
(cid:6) (cid:7)
x+y h(x)+h(y)
h ≤ 1+ .
2 2
which are the approximately midpoint convex function, (sometimes
calledtheapproximatelyJensenconvexfunctions)whichalsohavebeen
studied by several authors.
We give a general theory of S-almost convex functions. In partic-
ular when S has at least one point that is not a vertex we construct
(Definition 1.17 and Theorem 1.22) a bounded S-almost convex func-
tion E∆n: ∆ → R such that if h: ∆ → R is bounded, S-almost
S n n
convex, and h(e ) ≤ 0 on the vertices of ∆ then h(x) ≤ E∆n(x) for
k n S
all x ∈ ∆ . Then the number κ (n) := sup E∆n(x) is the sharp
n S x∈∆n S
constant in stability theorems of Hypers-Ulam type and the function
E∆m is the function that shows it is sharp (See Theorem 1.26.)
S
Probably the most natural choice for S are S = ∆ a simplex and
m
S = {(1/(m+1),...,1/(m+1))} the barycenter of a simplex. In these
cases we are able to give very explicit formulas both for the extremal
function E∆n and for the κ (n) = sup E∆n(x). (For the case
S S x∈∆n S
S = ∆ this was done in our earlier paper [3]. For the case of S the
m
barycenter of ∆ see Theorems 3.1.) There is an interesting dichotomy
m
in these two cases. When S = ∆ then E∆n is a concave piecewise
m S
linear function that is continuous on the interior ∆◦ of ∆ and the
n n
maximum occurs at the barycenter of ∆ . (See [3].) However when
n
S = {(1/(m + 1),...,1/(m + 1))} is the barycenter of ∆ then E∆n
m S
is discontinuous on a dense subset of ∆ and the graph of E∆n is a
n S
fractal with a large number of self similarities. See Figure 2.
This paper is not completely self-contained. Several of the results
have proofs that are very similar to the proofs in our earlier paper [2]
and at several places we refer the reader to [2] for proofs.
(cid:1)1.1. Definition and basic properties. Let ∆m := {(t0,...,tm) :
m t = 1,t ≥ 0} be the standard m-dimensional simplex. For the
k=0 k k
rest of this section we fix a subset
(cid:8)∞
S ⊂ ∆ .
m
m=1
ItfollowseasilyfromthedefinitionofS-almostconvexthatAlmCon (C)
S
is a convex subset of the vector space of all functions form C to R.
4 DILWORTH, HOWARD, AND ROBERTS
It is useful to make a distinction between two cases:
(cid:2)
1.2. Definition. If S ⊆ ∞ ∆ then
(cid:2) m=1 m
(1) If S (cid:2) N ∆ for any finite N then S is of infinite type.
(cid:2)m=1 m
(2) If S ⊆ N ∆ for some N then S is of finite type. If further
m=1 m
S ⊆ ∆ for some m then S is homogeneous. (cid:1)
m
(cid:2)
∞
1th.3e.nRaetmuraarlkt.oIpfowloegayss(uUm⊆e t(cid:2)h∞at th∆e unisioonpenmi=ff1U∆m∩∆is diissjooinptenanind∆has
m=1 m m m
for all m) then it is not hard(cid:2)to see that S is of finite type if and only
if it has compact closure in ∞ ∆ . (cid:1)
m=1 m
When considering S-almost convex functions there is no real distinc-
tion between S of finite type and S homogeneous.
(cid:2)
1.4. Proposition. Let S ⊆ N ∆ . For m ≤ N let ιm: ∆ → ∆
m=1 m N m N
be the inclusion ιn(t ,...,t ) = (t ,...,t ,0,...,0) and set S∗ =
N 0 m (cid:2) 0 m m
ιm[S ∩ ∆ ] ⊆ ∆ . Let S∗ = N S∗ ⊆ ∆ . Then for any convex
N m N m=1 m N
subset C of a real vector space AlmCon (C) = AlmCon (C).
S∗ S
Proof. This is a more or less straightforward chase though the defini-
tion. (cid:1)
The proof of the following is also straightforward and left to the
reader.
1.5. Proposition. Let S ⊆ ∆ and let
m
(cid:8)
S∗ = {(t ,t ,...,t ) : (t ,t ,...,t ) ∈ S}
ρ(0) ρ(1) ρ(m) 0 1 m
ρ∈sym(m+1)
where sym(m + 1) is the group of all permutations of {0,1,...,m}.
Then for any convex subset C of a real vector space AlmCon (C) =
S∗
AlmCon (C).
S
The following is also trivial.
(cid:2)
1.6. Proposition. Let S ⊆ S ⊆ ∞ ∆ . Then for any convex
1 2 m=1 m
subset C of a real vector space AlmCon (C) ⊆ AlmCon (C). (cid:1)
S S
2 1
ThefollowingcanbeusedtoreducecertainquestionsaboutS-almost
convex functions to the case where S ⊆ ∆ .
1
(cid:2)
1.7. Proposition. Let S ⊆ ∞ ∆ and let S be a nonempty sub-
m=1 m 1
set of S ∩ ∆ for some m. Let N ,...,N be a partition of the set
m 0 k
{0,1,...,m} into k +1 nonempty sets and let
S := {(α (t),α (t),...,α (t)) : t ∈ S } ⊆ ∆
2 0 2 k 1 k
ALMOST CONVEX FUNCTIONS 5
where (cid:4)
α (t) := t .
j i
i∈N
j
Then
AlmCon (C) ⊆ AlmCon (C)
S S
2
for any convex subset C of a real vector space. In particular if
(t ,...,t ) ∈ S and for some k ∈ {1,...,m−1} we set α = t +···+t
0 m 1 k
and β = t +···+t then any S almost convex function h will satisfy
k+1 m
h(αx )+h(βx ) ≤ 1+αh(x )+βh(x ).
0 1 0 1
Proof. Let C be a convex subset of a real vector space and let
y ,...,y ∈ C, α ∈ S and h ∈ AlmCon (C). Let x ,...,x ∈ C be
0 k 2 S 0 m
defined by
x = y if i ∈ N
i j j (cid:1)
As α ∈ S there is a t = (t ,...,t ) ∈ S ⊆ S so that α = t .
2 0 m 1 j i∈N i
j
Then as h is S-almost convex
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:4)k (cid:4)m (cid:4)k (cid:4)m
h α y = h t x ≤ 1+ t h(x ) = 1+ α h(y ).
j j i i i i j j
j=0 i=0 i=0 i=0
Thus h ∈ AlmCon (C). (cid:1)
S
2
It is useful to understand when an S-almost convex function is
bounded.
(cid:2)
1.8. Theorem. Let S ⊆ ∞ ∆ and assume that S contains at least
m=1 m
one point that is not a vertex (that is there is (t ,...,t ) ∈ S with
0 m
max t < 1). Let U be a convex open set in Rn. Then any S-almost
i i
convex function h: U → R which is Lebesgue measurable is bounded
above and below on any compact subset of U.
Proof. Let (t ,...,t ) ∈ S with max t < 1. Then there is a k ∈
0 m i i
{1,...,m−1} so that if α = t +···+t and β = t +···+t , then
1 k k+1 m
0 < α,β < 1, α+β = 1 and by Proposition 1.7
h(αx +βx ) ≤ 1+αh(x )+βh(x ).
0 1 0 1
We assume that α ≤ β, the case of α > β having a similar proof. As
any compact subset of U is contained in a bounded convex open subset
of U we can also assume, without loss of generality, that U is bounded.
Let K ⊂ U be compact and let r = dist(K,∂U). For any x ∈ Rn let
B (x) be the open ball of radius r about x. Then for any a ∈ K we
r
have B (a) ⊆ U. For a ∈ K define θ : Rn → Rn by
r a
1 α
θ (x) = a− x.
a
β β
6 DILWORTH, HOWARD, AND ROBERTS
Then it is easy to check that θ (a) = a for all a ∈ Rn and αx +
a
βθ (x) = a for all x ∈ Rn. Also θ is a dilation in the sense that
a a
(cid:13)θ (x )−θ (x )(cid:13) = (α/β)(cid:13)x −x (cid:13) for all x ,x ∈ Rn. As θ (a) = a
a 1 a 0 1 0 0 1 a
and (α/β) ≤ 1 this implies θ [B (r)] = B ((α/β)r) ⊆ B (r). Let Ln
a a a a
be Lebesgue measure on Rn. Then for any measurable subset P of Rn
Ln(θ [P]) = (α/β)nLn(P).
a
Choose a positive real number ε so that
(cid:6) (cid:6) (cid:7) (cid:7) (cid:6) (cid:7)
n n
α α
(1.2) 1+ ε < Ln(B(r))
β β
where B(r) is the open ball of radius r about the origin. Because h is
measurable and Ln(U) < ∞ there is a positive M so large that
Ln{x ∈ U : h(x) > M} < ε.
Therefore if V := {x ∈ U : h(x) ≤ M} then Ln(U (cid:3) V) < ε. Let
A := B (r) ∩ V. We now claim that A ∩ θ [A] has positive measure.
a a
For if not then A and θ [A] would be essentially disjoint subsets of
a
B (a) and therefore, using that Ln(θ [A]) = (α/β)nLn(A),
r a
Ln(B (r)) ≥ Ln(A)+Ln(θ [A])
a a
(cid:6) (cid:6) (cid:7) (cid:7)
n
α
= 1+ Ln(A)
β
(cid:6) (cid:6) (cid:7) (cid:7)
n
α
≥ 1+ (Ln(B (r))−ε)
a
β
which can be rearranged as (1+(α/β)n)ε ≥ (α/β)nLn(B(r)) contra-
dicting (1.2). Therefore Ln(A ∩ θ [A]) > 0 as claimed. Let a (cid:3)= x ∈
a
A∩θ [A]). Then x and θ (x) are both in A = B (r)∩V and therefore
a a a
h(x),h(θ (x)) ≤ M. Thus
a
h(a) = h(αx+βθ (x)) ≤ 1+αh(x)+βh(θ (x)) ≤ 1+αM+βM = M+1
a a
which shows that h is bounded above on K.
ToshowthathhasalowerboundoncompactsubsetsofU, leta ∈ U
and let r > 0 be small enough that the closed ball B (r) is contained
a
in U. Then B (r)is compact so by what we have just done there is a
a
constant C > 0 so that h(x) ≤ C for all x ∈ B (r). Let x ∈ B (r).
a a
Then, again as above, θ (x) ∈ B (r), and therefore
a a
h(a) = h(αx+βθ (x)) ≤ 1+αh(x)+βh(θ (x)) ≤ 1+αh(x)+βC
a a
which can be solved for h(x) to give
1
h(x) ≥ (h(a)−1−βC).
α
ALMOST CONVEX FUNCTIONS 7
Therefore h is bounded below on B (r). But any compact subset of
a
U can be covered by a finite number of such open balls and thus h is
bounded below on all compact subsets of U. (cid:1)
The following will be needed later.
1.9. Corollary. Let h: [a,b] → R be a Lebesgue measurable function so
that h(αx+βy) ≤ 1+αh(x)+βh(y) for some α,β > 0 with α+β = 1
(that is h is S-almost convex with S = {(α,β)} ⊂ ∆ ). Then h is
1
bounded above on [a,b].
Proof. By doing a linear change of variable (which preserves S-almost
convexity) we can assume that [a,b] = [0,1]. Also by replacing h by
x (cid:15)→ h(x)−((1−x)h(0)+xh(1)) we can assume that h(0) = h(1) = 0.
Let δ = α/(1+α). Then by Theorem 1.8 there is a constant C > 0
1
such that h(x) ≤ C on [δ,1−δ]. Let
1
C = max{C ,1/(1−α)+αC }.
2 1 1
We now show that h ≤ C on [0,1]. If x = 0, x = 1, or x ∈ [δ,1−δ] this
2
is clear. Let x ∈ (0,δ) then the choice δ is so that there is a y ∈ [δ,1−δ]
such that x = αky for some positive integer k. Also, as y ∈ [δ,1−δ],
h(y) ≤ C . Therefore
1
h(x) = h(αky) = h(β0+ααk−1y)
≤ 1+βh(0)+αh(αk−1y) = 1+αh(αk−1y)
(cid:9) (cid:10)
≤ 1+α 1+αh(αk−2y) = 1+α+α2h(αk−2y)
≤ 1+α+α2 +···+αk−1 +αkh(y)
1
≤ +αC ≤ C .
1−α 1 2
If x ∈ (1−δ,1) a similar calculation shows that h(x) ≤ C (or this can
2
be reduced to the case x ∈ (0,δ) by the change of variable x (cid:15)→ (1−x)).
This completes the proof. (cid:1)
1.2. A general construction for the extremal S almost convex
set on a simplex. We will show that on the n-dimensional simplex
∆ there is a pointwise largest bounded S-almost convex function that
n
vanishes on the vertices of ∆ . We start with some definitions.
m
1.10. Definition. A tree, T, is a collection of points N, called nodes,
and a set of (directed) edges connecting some pairs of (cid:2)nodes with the
following properties: The set N is a disjoint union N = ∞ N where
k=0 k
N contains exactly one point, the root of the tree, each N is a finite
0 k
set and if N = {v ,...,v } then N is a disjoint union N =
k 1 m m+1 m+1
8 DILWORTH, HOWARD, AND ROBERTS
P ∪···∪P of nonempty sets where P is the set of successors of
1 m i
v . The (directed) edges of the tree leave a node and connect it to its
i
successors and there are no other edges in the tree (cf. Figure 1). If v
is a node of the tree then r(v) := k where v ∈ N is the rank of v. A
k
branch of the tree is a sequence of nodes (cid:17)v (cid:18)∞ where v is the root,
k k=0 0
r(v ) = k, and there is an edge from v to v . (cid:1)
k k k+1
We now consider trees with extra structure, a labeling of the edges
in a way that will be used in defining the extremal S-almost convex
function.
(cid:2)
1.11. Definition. LetS ⊆ ∞ ∆ benonempty. ThenanS-ranked
m=1 m
tree is a tree T with its edges labeled by non-negative real numbers
in such a way that for any node v of the tree there is an element
t = (t ,...,t ) ∈ S so that there are exactly m + 1 edges leaving v
0 m
and these are labeled by t ,...,t . The number t is the weight of
0 m i
the edge it labels. Figure 1 shows a typical S-ranked tree. (cid:1)
(cid:1)
The root := unique node of rank 0.
(cid:1)(cid:2)
(cid:1) (cid:2)
t t
0(cid:1) (cid:2)1
(cid:1) (cid:2)
(cid:1)(cid:1) (cid:2)(cid:1)
The rank one nodes.
(cid:3)(cid:3)(cid:4)(cid:4) (cid:5)(cid:6)
(cid:3) (cid:4) (cid:5) (cid:6)
s (cid:3) (cid:4)s r (cid:5) r (cid:6) r
0 1 0 2 1
(cid:3) (cid:4) (cid:5) (cid:6)
(cid:3)(cid:1) (cid:4)(cid:1) (cid:1)(cid:5) (cid:1) (cid:6)(cid:1)
The rank two nodes.
Figure 1. An S ranked tree showing the labeling of the edges
out of the root by t = (t0,t1) ∈ S and the edges out of the rank
one nodes by s = (s0,s1) ∈ S and r = (r0,r1,r2) ∈ S. In our
definition each node will have at least two edges leaving it and the
sum of the weights t0,...,tm of the weights of all edges leaving a
node is unity (as (t0,...,tm) ∈ ∆m). Finally, in the definition of
tree used here, all branches are of infinite length.
WenowdescribehowanS-rankedtreedeterminesaprobabilitymea-
sure on the set of branches of the tree. Let T be an S-ranked tree and
let X = X(T) be the set of all branches of T. If (cid:17)v (cid:18)∞ ,(cid:17)w (cid:18)∞ ∈ X
k k=0 k k=0
are two elements of X we can define a distance between them as
d((cid:17)v (cid:18)∞ ,(cid:17)w (cid:18)∞ ) = 2−(cid:2) where (cid:9) is the smallest index with v (cid:3)= w
k k=0 k k=0 (cid:2) (cid:2)
(and d((cid:17)v (cid:18)∞ ,(cid:17)w (cid:18)∞ ) = 0 if (cid:17)v (cid:18)∞ = (cid:17)w (cid:18)∞ ). While we will not
k k=0 k k=0 k k=0 k k=0