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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.06245 |
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Table of Contents:
- We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ encloses a volume of at least $\frac{4}{3}π$. The appendix presents an example illustrating an alternative aspect for this problem.