﻿ Relationships Involving the "Volume" of a Region, the "Area" of its Boundary Surface, and the Average Distance from an Interior Point to that Boundary
San José State University

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Thayer Watkins
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Relationships Involving the "Volume" of a Region,
the "Area" of its Boundary Surface, and the
Average Distance from an Interior Point
to that Boundary

There are some interesting relationships between regions and their boundaries that extend over several dimensions.

## Two Dimensions

Let A be a convex two dimensional regional surrounding the origin of polar coordinates and C its perimeter. Suppose the coordinates of the perimeter are given by r=R(θ).

The area A of A can be considered to be made up of the areas of a set of triangles. They are not necessarily isoceles triangles. One side is of length R(θ) and the other longer by the amount dR=(dR/dθ)dθ. The average of the two edges is then R(θ)+½((dR/dθ)dθ. Each triangle has a base width of R(θ)dθ. The area of each triangle is then

#### ½[ R(θ)+½((dR/dθ)dθ]R(θ)dθ but the product of infinitesimals is insignificant compared to terms involving only dθ and may be dropped

So the area of each triangle is

#### ½R(θ)·R(θ)dθ

Thus the total area is

#### A = ∫02π½R(θ)·R(θ)dθ A = ½∫02πR(θ)·R(θ)dθ

The circumference C of the perimeter is given by summation of the lengths of the edges along the perimeter; i.e.,

#### C = ∫02πR(θ)dθ

The Mean Value Theorem says that

#### ∫f(x)g(x)dx = f(x)∫g(x)dx

where x is an average of x over the interval of integration.

Then by the Mean Value Theorem

#### A = ½RC

where R is the average distance from the origin to the perimeter.

That distance is equal to the radius of the circle having the same area as A.

Without reference to the Mean Value Theorem R may just as well be defined as

#### R = 2A/C

A may be moved so that any of its interior point coincides with the origin. Therefore the average distance from any interior point of A to its perimeter is the same for all interior points..

## Three Dimensions

Let V be a convex three dimensional region surrounding the origin of a spherical coordinate system (r, θ, φ). Let S be its suface. Let this surface be defined by R(θ, φ). The angle θ ranges from 0 to 2π radians and φ ranges from 0 to π radians.

V may be considered to be composed of pyramids with apex points at the origin. The base of each pyramid is a square of dimensions Rdθ by Rdφ. The volume of such a pyramid is

#### dV = ⅓R· Rdθ·Rdφ = ⅓R³dθdφ

The total volume is therefore

#### V = ∫0π∫02π⅓R³dθdφ V = ⅓∫0π∫02πR·R²dθdφ

The area S of the surface is

#### S = ∫0π∫02πR²dθdφ

This means that by the Mean Value Theorem

#### V = ⅓RS

where R is the average radius of the surface S. Its value is equal to the radius of the sphere with the same volume as V.

Since V can be moved so that any of its interior points coincide with the origin it means that the average distance from any interior point of V to its surface S is the same.

## One Dimension

Although the analysis for this case is trivial it is satisfying to see that it exactly matches the analysis in the cases of the other two dimensions.

Let L be a line stretching from a to b, where a<0<b. Its length L is given by

#### L = ∫abdx = b − a

Its "surface" is the two end points. The size of this "surface" is 2. The average distance x from the origin to the "surface" points of the line is

#### x = (b + (-a))/2 = L/2

This is just the "volume" L divided by the "size" of its surface.

## Conclusions

For a convex region U of dimension D the average distance R from an interior point to its (D-1)-dimensional surface is

#### R = (1/D)VD/SD-1

where VD is the D dimesional volume of U and SD-1 is the (D-1) dimesional area of the boundary surface of U. R is the same for all interior points of U.