I have 4 balls which diameters are $26, 24, 13$ and $9$ cm. What is the minimum length $L$ of a closed parallelepiped in which I can put the 4 balls? In the final answer please call $L, l$ and $W$ the 3 sides of the parallelepiped, with $ L \geq l \geq W $
When looking for references please do not look at the following (1) publications that look for solutions providing maximum density (this is not the objective function here); (2) publications dealing with balls of equal sizes; (3) publications on "stacking" (packing ≠ stacking).
I have looked at the following references without being able to find either the solution, or at least a good local minimum.
Stoyan, Y.G., Scheithauer, G. & Yaskov, G.N. Packing Unequal Spheres into Various Containers. Cybern Syst Anal 52, 419–426 (2016). https://doi.org/10.1007/s10559-016-9842-1
Labib Yousef, Contribution à la résolution des problèmes de placement en trois dimensions, Doctorate University de Picardie Jules Verne (France), 2017, 172 pages. Downloadable from: https://www.theses.fr/2017AMIE0020.pdf
On Google Scholar, the request < Packing "unequal spheres" pdf > gives other references, almost all from 2016-2023 when you add to the request several names of researchers who made important contributions to the field (Hifi Stoyan Sutou Yousef).
You can fit them in a cube with side length $L=39.434$ by placing the centers as follows: \begin{matrix} i & x_i & y_i & z_i \\ \hline 1 &26.4338 &26.4338 &13.0000 \\ 2 &12.0000 &12.0000 &27.4338 \\ 3 &6.5011 &32.9331 &6.5000 \\ 4 &34.9337 &4.5007 &34.9337 \end{matrix}
Given the radii $r_i$, the problem is to minimize $L$ subject to \begin{align} r_i \le x_i &\le L - r_i &&\text{for all $i$} \\ r_i \le y_i &\le L - r_i &&\text{for all $i$} \\ r_i \le z_i &\le L - r_i &&\text{for all $i$} \\ (x_i - x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2 &\ge (r_i + r_j)^2 &&\text{for $i < j$} \end{align}