Determine the general value of c(d,n,n)

Determine the value of c(d,n,n), the minimum over all n-tuples of unit vectors in the Euclidean space R^d of the maximum Euclidean norm of their signed sum, for general dimensions d and integers n; in particular, determine c(d,d+1,d+1) for d≥3.

Background

The paper studies the quantity c(d,n,k), defined as the minimum, over n unit vectors in Rd, of the largest norm attained by a signed sum of k selected vectors. Prior work established several sharp results but did not determine the case k=n in general. The paper resolves the planar case c(2,n,n), while the corresponding higher-dimensional problem remains unresolved.

The authors emphasize that even the apparently comparatively specific value c(d,d+1,d+1) is unknown for dimensions d≥3, and refer to this as a conjectural problem from the cited literature. The unresolved general determination of c(d,n,n) is also connected to extending the paper’s circumradius results to higher-dimensional Minkowski sums of convex bodies.

References

The results in [2] leave open the very natural problem of computing c(d, n, n). This appears to be a difficult task, as even c(d, d + 1, d + 1) is unknown for d ≥ 3 (see [2, Conjecture 1]).

— A Sharp Bound on Large Planar Signed Vector Sums  (2502.13752 - Grundbacher, 19 Feb 2025) in Section 1, Introduction, p. 1

For $m\ne n+1$ signals the optimal configuration is unknown in general, the rank-constrained problem posed in included.

— The Equality Cases of the Weak Simplex Conjecture  (2608.19093 - Su et al., 19 Aug 2026) in Section Conclusion