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.

The second result concerns convex domains of finite inradius [[lambda].

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].

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

For any p [member of] (0, + [infinity]) and n-dimensional convex domains [OMEGA] of finite inradius 60 there are the following sharp inequalities

For any n-dimensional convex domain [OMEGA] of finite inradius [[delta].

If [OMEGA] is an n-dimensional convex domain of finite inradius, p and q are positive numbers, then for any f [member of] [H.

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.

d] and it has inradius r = 1/d and circumradius R = 1/[square root of d].

Z has circumradius R = [square root of 4+2 [square root of 2/4] and inradius r = 1+ [square root 2])/4.

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.