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Interior Sphere Condition

The interior sphere condition of radius r>0 for a closed set S subset= R^n requires that every boundary point x lie on the boundary of a closed ball of radius r contained in S (Nour et al. 2011). The ball may depend on x. This is a condition at the boundary, not a requirement that every point of S belong to a contained closed ball of radius r.

Every nonempty set that is closed and satisfies this condition is a union of contained closed balls of radius r/2 (Nour et al. 2011). In the plane, the radius can be increased to r/sqrt(3), sharply (Nour and Takche 2026). Nour (2026) reports that r/2 is optimal in each fixed dimension n>=3, even for compact sets. For every epsilon>0, his construction gives such a compact set containing a point that lies in no contained closed ball of radius at least (1/2+epsilon)r.

See also

Boundary, Closed Ball

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References

Nour, C. "A Counterexample to the Strong Version of the Union of Uniform Closed Balls Conjecture in Dimensions Three and Higher." 10 Nov 2026. https://arxiv.org/abs/2611.14181v1.Nour, C.; Stern, R. J.; and Takche, J. "Validity of the Union of Uniform Closed Balls Conjecture." J. Convex Anal. 18, 589-600, 2011. https://doi.org/10.68381/jca18037.Nour, C. and Takche, J. "Validity of the Strong Version of the Union of Uniform Closed Balls Conjecture in the Plane." 8 Mar 2026. https://arxiv.org/abs/2603.07588v1.

Cite this as:

Weisstein, Eric W. "Interior Sphere Condition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InteriorSphereCondition.html

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