Given a uniform planar convex region C, let us consider 2 centers of mass - the center of mass of the region as a whole and the center of mass of its boundary alone (assuming its boundary to have constant linear density). It is obvious that for all regular polygonal regions, both centers coincide. Even for other shapes such as the rectangle or ellipse or an equilateral triangle with all 3 vertices chopped off equally (it is now a hexagon), the two centers coincide.


**Question:** Are there uniform planar convex regions which have no rotational or reflection symmetry at all but with both centers of mass coincident? If so, how can we characterize regions where both centers of mass are coincident?


**Note:** This post continues [Polyhedrons and their centers of mass](https://mathoverflow.net/questions/403778/polyhedrons-and-their-centers-of-mass). Based on this earlier post, one can think of other centers of mass and also 3D analogs to this question.



When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide?
In Euclidean triangle geometry the Spieker point $S$ is the centroid of the triangle sides, and it is known to be the
anticomplement of the incircle center $I$ (<https://faculty.evansville.edu/ck6/encyclopedia/ETC.html>), which means it is in line with
$I$ and the triangle centroid $C$, such that
$$\overline{S C}~=~\frac{1}{2} \,\overline{I C}.$$
The following theorem generalizes this fact to $n$ polyhedra with inscribed spheres by a proof that is based on rescaling.


Definition:


The Spieker point of a $n$-polyhedron is the centroid of all its $n-1$ faces.


Theorem:


In every $n$-polyhedron $P$ with inscribed sphere, the incenter $I$, the centroid $C$, and the Spieker point $S$ lie in this order on a line, such that
$$\overline{S C}~=~\frac{1}{n} \,\overline{I C}.$$


Proof:


Denote by $v$ the $n$-volume, and by $a$ the area of the $n-1$-faces, and by $r$ the radius of the inscribed sphere of $P$.
For $\epsilon>0$, expand $P$ by a factor $1+\epsilon/r$ and translate it to $P\_\epsilon$ such that the insphere centers of $P$ and $P\_\epsilon$ coincide. Because the insphere radii are perpendicular to the faces in the touching points, the corresponding faces of $P$ and $P\_\epsilon$ are parallel and have distance $\epsilon$. The volume of $P\_\epsilon\backslash P$ is $ a\_\epsilon ~=~\epsilon\, a +O(\epsilon^2)$, its centroid is $S\_\epsilon$ and converges to $S$.  

By the expansion around $I$, the centroid $C\_\epsilon$ of $P\_\epsilon$ lies on a line with $I$ and $C$ and  

$$\overline{I C\_\epsilon}~=~(1+\epsilon/r) \,\overline{I C}, $$
such that
$$\overline{C C\_\epsilon}~=~\epsilon/r \,\overline{I C}. $$
On the other hand, $C\_\epsilon$ is the centroid of $C$ with weight $v$ and $S\_\epsilon$ with weight $ a\_\epsilon$, whereby $C,C\_\epsilon,$ and $S\_\epsilon$ lie on a line which then also contains $I$ and
$$v\,\overline{C C\_\epsilon}~=~\frac{v\,\epsilon}{r}\,\overline{I C}~=~a\_\epsilon\, \overline{S\_\epsilon C\_\epsilon}. $$
Taking the limit $\epsilon \to 0$  

results in
$$\overline{S C}~=~\frac{v}{a\,r} \,\overline{I C}.$$
Observing that the volume of the $n$-pyramid over any $n-1$ face $F$
of $P$ with tip $I$ is
$$ v(pyramid) ~=~\frac1n\,r\,area(F),$$
and that the sum $\frac1n\,a\,r$ of all these volumes is equal to the total volume $v$ of $P$, it follows that
$$\overline{S C}~=~\frac{v}{a\,r} \,\overline{I C}~=~\frac{1}{n} \,\overline{I C},$$
which
proves the theorem.


A corollary of this theorem is that in a $n$-polyhedron $P$ with inscribed sphere with centroid $\vec{c}$ and Spieker point $\vec{s}$ the incenter
is given by
$$\vec{i} ~=~ (n+1)\, \vec{c} - n\, \vec{s}.$$


As a consequence, centroid and Spieker point coincide in $n$-polyhedra with inscribed sphere if and only if incenter and centroid agree.


