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

,