MathDB

In the

(x,y)-

plane, if

R

is the set of points inside and on a convex polygon, let

D(x,y)

be the distance from

(x,y)

to the nearest point of

R.

(a) Show that there exists constants

a,b,c,

independent of

R

, such that

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x,y)} dxdy =a+bL+cA,

where

L

is the perimeter of

R

and

A

is the area of

R.

(b) Find the values of

a,b

and

c.

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