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arXiv · 2102.01787

A negative answer to Ulam's Problem 19 from the Scottish Book

Dmitry Ryabogin
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Abstract

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking {\it is a solid of uniform density which will float in water in every position a sphere?} Assuming that the density of water is $1$, we show that there exists a strictly convex body of revolution $K\subset {\mathbb R^3}$ of uniform density $\frac{1}{2}$, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions $d\ge 3$.

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Dmitry Ryabogin. 2021-02-02. A negative answer to Ulam's Problem 19 from the Scottish Book. https://arxiv.org/abs/2102.01787

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