[A.sub.n+1] be an n-dimensional simples in [R.sup.n] and r be the

inradius of A.

Recall that [r.sub.1] = qq' is the

inradius. In the three children this has changed to pq', qp', pp'.

where r(B) denotes the inradius of B and [Q.sub.l] is any square of side l.

Let us define: dB as the boundary of B, p(B) as its perimeter, d(B) as its diameter, r(B) as its inradius and [Q.sub.s] (x) as a square of side s centered at x.

The second result concerns convex domains of finite inradius [[lambda].sub.0] defined as

Namely, for any convex domain n of finite inradius 6q it was proved that

We will fix h [greater than or equal to] 0 and consider [lambda] as the constant best possible in (6) for the set of all n-dimensional convex domains with fixed inradius [[delta].sub.0].

If [OMEGA] is an n- dimensional convex domain of finite inradius 60, then the sharp inequality

We will show in Lemma 1 that the relative error [[??].sup.d.sub.k] (A)/S(A) is in a sharp way bounded from below by (a multiple of) the inradius of Z and from above by (a multiple of) the outer radius of Z.

and the corresponding zonotope [Z.sub.V] has inradius [r.sub.V] [approximately equal to] 0.18290 and circumradius [R.sub.V] [approximately equal to] 0.98199; see Fig.

leading to the inradius [??][approximately equal to] 0.191412 and circumradius [??][approximately equal to] 0.981147, respectively.

For example, he introduced the letter e to represent the base of the system of natural logarithms; the use of the Greek letter [pi] for the ratio of circumference to diameter in a circle also is largely due to Euler; the symbol i for [square root of -1]; the use of the small letters a, b and c for the sides of a triangle and of the corresponding capitals A, B and C for the opposite angles stems from Euler, as does the application of the letters r, R and s for the

inradius, circumradius and semiperimeter of the triangle respectively; the designation lx for logarithm of x; the use of [SIGMA] to indicate a summation; and, perhaps most important of all, the notation f(x) for a function of x are all due to Euler.