—   region (A bounded Euclidean region with positive finite intrinsic measure and a specified size. Uniformity is with respect to length, area or volume in the region, or arc length or surface area on a circle or sphere.)

$M(aK+b)=aM(K)$ for $a>0$ and a translation vector $b$, with the corresponding uniform measure on the transformed region.

For a regular $n$-gon with $n\geq3$, inradius $h$ and $\theta=\pi/n$, $M(K)=\frac h3\bigl[\sec\theta+\cot\theta\log(\sec\theta+\tan\theta)\bigr]$. If the circumradius is $R$, then $h=R\cos\theta$; if the side length is $a$, then $h=(a/2)\cot\theta$ [1].

For a rectangle of side lengths $a,b>0$, put $d=\sqrt{a^2+b^2}$. Then $M(K)=[2abd+a^3\log((b+d)/a)+b^3\log((a+d)/b)]/(12ab)$ [1].

For the solid ball $B_R^d\subset\mathbb R^d$ of radius $R$, with $d\geq1$, the radial density is $d r^{d-1}/R^d$ on $0\leq r\leq R$, so $M(B_R^d)=dR/(d+1)$.

For the sphere $S_R^{d-1}=\{x\in\mathbb R^d:\|x\|_2=R\}$ with uniform surface measure, $M(S_R^{d-1})=R$.

The interval has length $1$. The square, cube and equilateral triangle have side $1$, and the rectangle has sides $1$ and $2$. The disk, solid ball, circle and sphere have radius $1$. The regular pentagon, hexagon, heptagon and octagon have circumradius $1$, as in the corresponding planar domains.

The interval, polygons, disk, cube and solid ball carry normalized Lebesgue measure in their dimensions. The circle carries normalized arc length, and the sphere carries normalized surface area. Their centroids are their geometric centers. On the circle and sphere the distance is measured in the ambient Euclidean space.

Mean distance between two uniform random points averages over two independent sampled endpoints. Here one endpoint is fixed at the centroid. Areas of regular polygons, volumes of unit balls and surface areas of unit spheres give the measures used to normalize these regions.

The interval, disk, solid-ball, circle and sphere means are stored exactly. Each remaining entry is a certified decimal enclosure, with an exact expression in its comment. All logarithms are natural.

Sage

# Mean distance to the centroid of a regular n-gon of circumradius R.
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField

def mean_regular_polygon(n, R=1, digits=100):
    if n < 3:
        raise ValueError("n must be at least 3")
    F = RealBallField(numberdb.bits(digits, losing=80))
    theta = F.pi()/n
    h = F(R)*theta.cos()
    return h/3*(1/theta.cos()
                +(1/theta.cos()+theta.tan()).log()/theta.tan())

print(mean_regular_polygon(6))

H. Li and X. Qiu, Moments of Distance from a Vertex to a Uniformly Distributed Random Point within Arbitrary Triangles, Mathematical Problems in Engineering 2016 (2016), Article 8371750, 10 pages. (doi)

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box integrals, Journal of Computational and Applied Mathematics 206 (2007), 196–208. (doi)

Li and Qiu: moments of distance in triangles and polygons

Bailey, Borwein and Crandall: Box integrals, author manuscript

Mean distance between two uniform random points of a region —   Both endpoints are sampled independently, instead of fixing one at the centroid.

Areas of regular polygons —   Areas used to normalize the uniform measure on regular polygons.

Volumes of unit balls —   Volumes used to normalize the uniform measure on disks and solid balls.

Surface areas of unit spheres —   Surface measures used to normalize the uniform measure on circles and spheres.

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds thirteen classical sampling benchmarks: the unit interval, unit square, $1\times2$ rectangle, unit-side equilateral triangle, unit-radius disk, regular polygons with five through eight sides and circumradius $1$, unit cube, unit-radius solid ball in $\mathbb R^3$, unit circle and unit sphere in $\mathbb R^3$; this selection covers regular polygons with three through eight sides in the stated normalizations and familiar low-dimensional boxes and round regions without imposing an aspect-ratio or dimension grid)

Sources of data: [1], [2]

How they were obtained:

The eight irrational entries are evaluated throughout in Arb ball arithmetic, using 80 guard bits beyond 100 significant digits. The largest measured ball radius is less than $4.3\times10^{-124}$, and every relative radius is less than $10^{-120}$. The five rational entries are returned exactly. The regular-polygon and rectangle closed forms are given in equation (23) and Examples 1–3 of [3]. The square and cube forms follow from equation (4) and Section 7 of [4].

more

Independent quadratures integrate the radial distance density for polygons and balls, polar coordinates for rectangles, and a face-pyramid decomposition for the cube. They are run at 130 and 170 decimal digits, after normalization and second-moment controls, and agree with every closed form to absolute tolerance $10^{-120}$. Scaling is checked for every region. These numerical quadratures are agreement checks; the certified error bounds come from Arb evaluation of the closed forms. Direct uniform sampling of 300000 points in each region also checks the centroid, second moment and mean. The attached checks can compare all independent values and certified balls with the intervals read from the database.