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 Rn for n ≥ 3 there are no reduced simplices, which complicates the search for candidates. In R3, the Heil body currently provides the best-known benchmark among rotationally symmetric bodies, but a complete solution remains open.

References

The same problem in higher dimensions remains open, as there are no reduced simplices in n for n\geq 3 (see ), therefore there are no really good candidates for the volume minimizing problems -- so far the best one in 3 is the so-called Heil body, which has a smaller volume than any rotationally symmetric body of the same minimal width.

— On the area of ordinary hyperbolic reduced polygons  (2403.11360 - Sagmeister, 2024) in Section 1 (Introduction)

Another long-standing open problem is to determine convex bodies of constant width with minimal volume, where the Meissner bodies are conjectured to be optimal for dimension $d=3$.

— Galerkin approximations to the space of convex bodies by polytopes in nondegenerate V-representation  (2608.26615 - Rieger, 27 Aug 2026) in Section 1, Introduction