the Cauchy formula

Below is a more detailed (yet still relatively accessible) derivation of the 3D Cauchy formula (also called the mean chord length formula) in integral geometry, which states:

$(L)_\circ \;=\; \frac{4\,V}{A},$

where

$(L)_\circ$ is the mean chord length of a convex body $K \subset \mathbb{R}^3$ ;
$V$ is the volume of $K$ ;
$A$ is the surface area of $K$ .

This formula can be found in classical references such as:

L. A. Santaló, Integral Geometry and Geometric Probability (Addison-Wesley, 1976).
D. G. Kendall & P. A. P. Moran, Geometrical Probability (Hafner, 1963).

Below, we give a step-by-step outline that highlights the key ideas and how one arrives at the factor $4$ .

1. The Setting: Random Chords in a Convex Body

Let $K \subset \mathbb{R}^3$ be a convex body—that is, a compact, convex set with nonempty interior. We want to define a “random chord” and then compute the expected length of such a chord.

1.1 How Do We Choose a “Random Chord”?

There are several equivalent ways to define a random chord in $K$ . The classical approach in integral geometry is:

Choose a random direction $\mathbf{u}$ uniformly on the unit sphere $S^2$ .
Choose a random parallel chord in that direction, with a uniform distribution over all parallel chords (i.e., over all lines in direction $\mathbf{u}$ that intersect $K$ ).

Concretely, if $\mathbf{u}$ is fixed, consider the family of planes perpendicular to $\mathbf{u}$ . You pick one such plane “at random” among those that intersect $K$ . The line of intersection with $K$ (in direction $\mathbf{u}$ ) is then your random chord.

1.2 The Mean Chord Length $(L)_\circ$

Once we have a chord, let us denote its length by $\ell$ . By mean chord length we mean the expectation

$(L)_\circ \;=\; \mathbb{E}[\ell].$

We will show

$(L)_\circ \;=\; \frac{4\,V}{A}.$

2. Ingredients from Integral Geometry

The rigorous proof relies on some foundational results of integral geometry (particularly Crofton-type and Cauchy-type formulas). We focus on the key geometric steps, omitting some of the heavier measure-theoretic justifications.

2.1 Chords, Slices, and Volume

Let us fix a direction $\mathbf{u}\in S^2$ . For any real number $t$ , consider the plane

$P(t,\mathbf{u}) \;=\; \{\,x \in \mathbb{R}^3 : \langle x, \mathbf{u}\rangle = t\},$

where $\langle \cdot,\cdot\rangle$ is the usual dot product. The intersection

$K_{t,\mathbf{u}} \;=\; K \,\cap\, P(t,\mathbf{u})$

is a (possibly empty) cross-sectional slice of $K$ . Then:

The volume $V$ of $K$ can be recovered by integrating the areas of the slices:
$V \;=\; \int_{-\infty}^{\infty} \bigl[\mathrm{Area}\,\bigl(K_{t,\mathbf{u}}\bigr)\bigr] \,dt.$
Chords parallel to $\mathbf{u}$ arise as intersections of $K$ with lines $L\|\mathbf{u}$ . The set of all such lines can be parametrized by $(t,y)$ where $t$ is as above and $y$ is a point in the plane $P(t,\mathbf{u})$ . In other words, each chord is determined by picking which plane it lies in (via $t$ ) and where in that plane it goes through (via $y$ ).

2.2 Relating Chord Lengths to Surface Area

Although volume slicing explains how chord lengths integrate to give volume, we also need a relation to the surface area $A$ . A central insight from integral geometry is that, when averaged over all directions, the measure of chords “touching” or “piercing” the boundary is proportional to $A$ .

In fact, there is a known relationship sometimes called the Cauchy–Crofton-type formula in 3D:

The total “measure” of all lines (in all directions) that intersect $K$ is related to $V$ .
The total “measure” of all lines (in all directions) that intersect the boundary $\partial K$ is related to $A$ .

One then exploits the interplay between these two measures (volume-based and area-based) to deduce the average chord length must be a constant multiple of $\tfrac{V}{A}$ . The crux is pinning down that constant as exactly $4$ .

3. Sketch of the Proof

We will outline a reasonably direct proof, following the spirit of “slicing” and “averaging” over directions, showing why the formula must be

$(L)_\circ \;=\; \frac{4\,V}{A}.$

3.1 Counting “Chord-Length $\times$ Direction” Two Ways

Pick a Direction $\mathbf{u}$
Let $\mathbf{u}$ be fixed for the moment. Consider all chords of $K$ parallel to $\mathbf{u}$ . You can parametrize each chord by its midpoint $m$ in the plane $P(t,\mathbf{u})$ . Geometrically, as $m$ sweeps out the cross-section $K_{t,\mathbf{u}}$ , the corresponding chord is the line in direction $\mathbf{u}$ through $m$ , truncated by the boundary of $K$ . Its length is $\ell\bigl(m,\mathbf{u}\bigr)$ .
Integrate the Chord Length Over the Cross-Section
If you “sum up” (i.e. integrate) $\ell(m,\mathbf{u})$ over all midpoints $m\in K_{t,\mathbf{u}}$ , and then integrate over all $t$ , you effectively recover a quantity proportional to the volume $V$ . (This is akin to slicing arguments in integral geometry: each small element of volume can be thought of as “contributing” to exactly one chord in each direction, in measure-theoretic sense.)
Average Over All Directions
After performing the above step for one fixed $\mathbf{u}$ , we then let $\mathbf{u}$ vary uniformly over the sphere $S^2$ . This “double integration” (over directions $\mathbf{u}$ and over chord-midpoints in each cross-section) can be reorganized to show a proportionality between the total chord length measure and the ratio $\tfrac{V}{A}$ . The boundary $\partial K$ appears naturally when one keeps track of how chords terminate on $\partial K$ .

3.2 Pinning Down the Constant = 4

What remains is to show that the constant of proportionality is 4, rather than some other number. One way to see why is to look at special shapes for which the formula can be computed independently—for instance, a sphere.

Example: The Unit Sphere

Consider $K$ as the unit sphere in $\mathbb{R}^3$ . Then:

$V = \tfrac{4}{3}\pi r^3$ with $r=1$ , so $V = \tfrac{4\pi}{3}$ .
$A = 4\pi r^2$ with $r=1$ , so $A = 4\pi$ .
Hence, $\tfrac{V}{A} = \tfrac{4\pi/3}{4\pi} = \tfrac{1}{3}$ .

Now, what is the mean chord length $(L)_\circ$ of a unit sphere if chords are chosen in the manner above?

A random direction $\mathbf{u}$ is irrelevant for a sphere (all directions are “the same”), so we just pick a random chord uniformly among all parallel chords.
For a sphere, all parallel chords are parallel “slices.” If you pick a random chord by uniformly picking its midpoint inside the circular cross-section, classical geometric probability shows that the midpoint is more likely to be near the center. The resulting mean chord length (in a unit sphere) turns out to be $\tfrac{4}{3}r = \tfrac{4}{3}$ when $r=1$ .

In other words, one can do a quick integral or use well-known results about chord distributions in a circle/sphere to see that the expected chord length in a unit sphere is $\tfrac{4}{3}$ . Comparing this with $\tfrac{V}{A} = \tfrac{1}{3}$ for $r=1$ , we get:

$(L)_\circ \;=\; 4 \times \frac{V}{A},$

confirming the constant must indeed be $4$ .

For any convex body $K$ , a more sophisticated integral-geometric argument (a “transport-of-measure” or “Crofton–Santaló approach”) generalizes exactly this result to show the constant remains $4$ .

4. Making It More Rigorous: Crofton–Santaló Formalism

A more rigorous treatment involves:

Measure of Lines in 3D
One defines a measure on the space of all lines in $\mathbb{R}^3$ that is invariant under rigid motions (translations + rotations). Concretely, this measure can be described by choosing $\mathbf{u}\in S^2$ (the direction) uniformly and then choosing the perpendicular distance from the origin in a certain way.
Counting the Lines That Intersect $K$
The integral of the “indicator function” $\mathbf{1}\{\text{line} \cap K \neq \emptyset\}$ with respect to this measure on lines is proportional to $V$ .
Counting the Lines That Hit the Boundary $\partial K$
Likewise, if one counts lines according to where and how they intersect $\partial K$ , one obtains an expression proportional to $A$ .
Relating Total Chord Length
When you want the sum (or integral) of chord lengths over all such random lines, you effectively do a more refined version of (2) and (3). The difference between the boundary-based counting and volume-based counting isolates a factor that emerges as exactly $4$ .

Putting it all together yields the final statement:

$\boxed{ (L)_\circ \;=\; \frac{4\,V}{A}. }$

5. Additional Perspective and Intuition

Dimensional Analysis
- $[V]$ has dimensions of $\text{(length)}^3$ .
- $[A]$ has dimensions of $\text{(length)}^2$ .
- Thus, $\tfrac{V}{A}$ has dimension of $\text{(length)}$ .
- If there is a universal constant relating a mean chord length to $V$ and $A$ , it must be dimensionless—and turns out to be $4$ in 3D.
Analogy with 2D
In 2D, the analogous formula says that the mean chord length of a convex domain (mean “random segment” through the region) is

$(L)_\circ \;=\; \frac{\pi\,(\text{Area})}{\text{Perimeter}}.$ The constant there is $\pi$ . In 3D, it becomes $4$ . One might have guessed a $\pi$ -related constant in 3D as well, but the integral geometry shows that $4$ emerges instead.
Special Cases
- Sphere (already seen): mean chord length $= \tfrac{4}{3}r$ .
- Cube or Rectangular Box: one can attempt a direct (though tedious) calculation or rely on the general theorem.

6. Concluding Remarks

The Cauchy mean chord length formula in 3D,

$\boxed{(L)_\circ = \frac{4V}{A}},$

is a cornerstone of integral geometry. It reveals a deep connection among:

The 1D quantity: average chord length,
The 2D boundary measure: $A$ ,
The 3D interior measure: $V$ .

While the final statement $\bigl((L)_\circ = 4V/A\bigr)$ looks simple, its proof showcases the power of geometric probability and the invariance properties of “random directions” and “random chords.” The sphere example clinches why the constant is specifically $4$ . For a fully rigorous derivation—with all measure-theoretic details—one consults the classic treatments in Santaló’s or Kendall & Moran’s texts, which systematically develop the necessary Crofton–Cauchy–Santaló formulas in multiple dimensions.