4 1. THE RESULT
e can be identified with an element of the unit sphere
Sd−1,
so
Sd−1
is the set of
all directions in
Rd.
Let K be a convex polytope, x K and e
Sd−1
a direction.
The line le,x through x which is parallel with e intersects K in a segment Ae,xBe,x.
We call the minimum of the distances between x and Ae,x,Be,x the distance from
x to the boundary of K in the direction of e:
(1.5) dK (e, x) = min{dist(x, Ae,x), dist(x, Be,x)},
while
(1.6)
˜
d
K
(e, x) = dist(x, Ae,x) · dist(x, Be,x)
could be called the normalized distance. Note that even if x lies on the boundary
of K, it may happen that dK (e, x),
˜
d K(e, x) 0; for example, if K is a cube of
side length a, x is the midpoint of an edge and e is the direction of that edge, then
dK (e, x) =
˜
d K(e, x) = a/2.
If f is a continuous function on K, then we define its r-th symmetric differences
in the direction of e as
(1.7) Δhef(x)
r
=
r
k=0
(−1)k
r
k
f x + (
r
2
−k)he
with the agreement that this is 0 if x +
r
2
he or x
r
2
he does not belong to K.
Finally, define the r-th modulus of smoothness as (see [12, Section 12.2])
(1.8) ωK
r
(f, δ) = sup
e∈Sd−1, h≤δ, x∈K
|Δr
h
˜
d
K
(e,x)e
f(x)|,
which we shall often write in the form
(1.9) ωK
r
(f, δ) = sup
e∈Sd−1
sup
h≤δ
Δr
h
˜K
d (e,x)e
f(x)
K
,
i.e. ωK
r
(f, δ) is the supremum of the directional moduli of smoothness
ωK,e(f,
r
δ) := sup
h≤δ
Δr
h
˜K
d (e,x)e
f(x)
K
for all directions. Note that when K = [−1, 1], then there is only one direction
(and its negative) and this modulus of smoothness takes the form (1.1), i.e.
(1.10) ωϕ(f,
r
δ) =
ω[r−1,1](f,
δ).
Another way to write the modulus of smoothness (1.8) is
(1.11) ωK
r
(f, δ) = sup
I
sup
h≤δ
Δr
h
˜
d
K
(e,x)e
f(x)
I
= sup
I
ωI
r(f,
δ),
where I runs through all chords of K, so ωK
r
(f, δ) is just the supremum of all the
moduli of smoothness ωI
r(f,
δ) on chords of K, and here ωI
r(f,
δ) is just the analogue
(actually a transformed form) of the ϕ-modulus of smoothness ωϕ
r
for the segment
I.
It is also immediate that
(1.12) ωϕ(f,
r
δ) ωϕ(f,
r
1), for all δ 1,
and as a consequence,
(1.13) ωK
r
(f, δ) ωK
r
(f, 1), for all δ 1.
We also set
En(f)K = inf
Pn
f Pn
K
,
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