Higher-dimensional Euclidean isominwidth problem

Establish, for Euclidean spaces R^n with n ≥ 3, the minimizers of volume among convex bodies with a fixed minimal width; that is, determine the minimal volume and characterize all convex bodies attaining it in the higher-dimensional isominwidth problem.

Background

Pal proved that in the Euclidean plane, among convex bodies with fixed minimal width, the regular triangle minimizes area. Extending this to higher dimensions leads to the higher-dimensional isominwidth problem.

The paper notes that in R^n for n ≥ 3 there are no reduced simplices, which complicates the search for candidates. In R^3, the Heil body currently provides the best-known benchmark among rotationally symmetric bodies, but a complete solution remains open.