MathDB

Let

C

be a fixed unit circle in the cartesian plane. For any convex polygon

P

, each of whose sides is tangent to

C

, let

N( P, h, k)

be the number of points common to

P

and the unit circle with center at

(h, k).

Let

H(P)

be the region of all points

(x, y)

for which

N(P, x, y) \geq 1

and

F(P)

be the area of

H(P).

Find the smallest number

u

with

\frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u

for all polygons

P

, where the double integral is taken over

H(P).

B6