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A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c). Such a convex ...

Let $K$ be a convex body in $\mathbb{R}^3$. Suppose $K$ is held at some position and orientation on an inclined plane, and released. Let there be sufficient friction so that it rolls without slippage. ...

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only one orientation of stable equilibrium and one orientation of unstable ...

The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one ...