In the Euclidean plane, given is a collection of $k$ circular disks $D_1,…,D_k$ of radii $r_1,…, r_k$, supplied with weights $w_1,…,w_k$, assuming that each circle’s center of gravity coincides with the circle’s center (in general, the weights are independent from the radii). If the circles are packed in a circular container $D$ so that the center of gravity of their configuration coincides with the center of the container, then we say that the packing is *balanced*, and it is called the *tightest balanced packing* if the radius of the container is the smallest possible. The natural problem is, for a given finite family of weighted circular disks, to determine its tightest balanced packing. Generalizations to other shapes and higher dimensions are natural.

**Question 1.** Does anyone know of any references on this subject?

**Some trivial inequalities.** Let $r_b$ denote the minimum "balanced" radius, while $r$ - the minimum packing radius, no weights. Obviously, $r\le r_b$. But obviously as well, $r_b\le 2r$. A bit less obviously, $r_b<2r$ while $r_b$ can be arbitrarily close to $2r$.

**Question 2.** The natural case in which the weight of a disk coincides with its area, resp. volume, is of special interest: how much in this case can the radius of the container for a tightest balanced packing differ from the radius in the tightest packing without the balance requirement? To be more precise: what is the least upper bound on the ratio ${r_b}\over{r}$?

(Of course, Wlodek knows all this).=== In the nondegenerated case, the inequality is sharp. $\endgroup$ – Wlod AA Jun 4 '18 at 23:58