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Grünbaum's Equal-Diameter Covering Problem

Grünbaum's equal-diameter covering problem asks for the least number g_d of closed balls of radius 1/2 sufficient to cover every bounded set of generalized diameter 1 in R^d (Grünbaum 1963). The proposed bound g_d<=d+1 fails in five dimensions (Arman et al. 2026).

Arman et al. (2026) constructed a compact set of generalized diameter 1 that requires at least seven closed balls of radius 1/2. The set consists of matrices (uu^T-I/3)/sqrt(2), where u ranges over the unit sphere in R^3 and I is the 3×3 identity matrix. The ambient space is the five-dimensional space of traceless real symmetric matrices with the Frobenius norm. The set is the Veronese projective plane scaled by 1/sqrt(3). They also found a finite diameter-one set in R^5 that cannot be covered by six closed balls of radius 0.50001. The compact example admits a set partition into six sets of smaller generalized diameter, as in Borsuk's conjecture.

See also

Borsuk's Conjecture, Closed Ball, Generalized Diameter

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References

Arman, A.; Bondarenko, A.; Prymak, A.; and Radchenko, D. "A Diameter-One Set in R^5 That Cannot Be Covered by Six Diameter-One Balls." 5 Nov 2026. https://arxiv.org/abs/2611.08198v1.Grünbaum, B. "Borsuk's Problem and Related Questions." In Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, Held at the University of Washington, Seattle, June 13-17, 1961 (Ed. V. Klee). Providence, RI: Amer. Math. Soc., pp. 271-284, 1963.

Cite this as:

Weisstein, Eric W. "Grünbaum's Equal-Diameter Covering Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrunbaumsEqual-DiameterCoveringProblem.html

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