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Does there exist a 4-dimensional object whose intersection with a hyperplane always produces a torus, regardless of the orientation of the hyperplane? I am assuming that the hyperplane passes through the origin.

If we try generalizing the problem, we can see that it is true in dimensions smaller than 4. Consider the solid shape in $\mathbb{R}^3$ formed by the inequality, $a \leq ||\small \vec v||_2 \leq b$, where $||\small \vec v||_2$ is the L$_2$ norm of $\vec v$. Intersecting this shape with a plane through the origin produces the 2-dimensional analog of a torus.

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