Isoperimetric problem with application to the figure of cells

Biological applications suggest the following geometrical problem. Consider n three-dimensional cells, touching or not, and assume that the free energy of their figure is the sum H = A + αB of the area A of the cell walls adjacent to the ambient fluid plus an adjustable constant 0 ≤ α ≤ 2 times the area B of the walls separating two cells. Given the partial volumes of the cells, the problem is to describe the shape of the (optimal) figure that renders H as small as possible; the analogous problem for two-dimensional cells is the subject of this paper. Geometrical proofs of the following features of optimal two-dimensional figures are presented below: (a) the edges bounding the cells are circular arcs; (b) at an inside corner, three edges meet at angles 2π/3; (c) at an outside corner, three edges meet with outside angle 2 cos^-1 α/2; (d) pressures can be ascribed to the cells so that the pressure drop across an edge is proportional to its curvature; (e) bubbles appear at each inside corner as α passes 3^1/2. All these facts have three-dimensional analogues with similar proofs.