Positions of leading tetrahedral-symmetric constant-width candidates in the minimal-volume landscape

Determine the relative volume ordering of, and ascertain whether any of, the following bodies achieves the minimal volume among all three-dimensional bodies of constant width that are invariant under the symmetry group of the regular tetrahedron: (i) the Minkowski average of the two Meissner bodies, (ii) the three-dimensional peabody constructed via focal conics, and (iii) the shadow body π(M) obtained by orthogonally projecting onto the hyperplane orthogonal to E the four-dimensional constant-width body M defined as the intersection of unit balls centered on selected subsets of the 2-skeleton of the Reuleaux 4-simplex.

Background

The paper constructs a new four-dimensional body M of constant width by modifying a Reuleaux 4-simplex and then studies its orthogonal projection π(M) onto the hyperplane orthogonal to the vertex vector E. This shadow π(M) is a three-dimensional body of constant width with full tetrahedral symmetry and distinctive elliptical edges.

In three dimensions, prominent constant-width bodies with tetrahedral symmetry include the Meissner bodies, their Minkowski averages, and the peabody derived from focal conics. The authors provide numerical evidence that π(M) has volume close to that of Meissner bodies and significantly less than the unit-width ball, motivating a comparative extremal problem within the tetrahedral-symmetric class.

The authors explicitly state that the standing of these candidates within the minimization landscape for volume under tetrahedral symmetry is unresolved, highlighting the need to determine their relative ordering and whether any attains the minimum.