On the curvature bounded sphere problem in $\mathbb{R}^3$

Abstract: 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 $\frac{4}{3}\pi$. The appendix presents an example illustrating an alternative aspect for this problem.