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A negative answer to Ulam’s problem 19 from the Scottish Book. (English) Zbl 1529.52008

In this paper, the author answers in the negative an old problem posed by Ulam in the Scottish Book (as Problem 19), who asked if the three-dimensional (Euclidean) ball is a unique convex body \(K\) in the three-dimensional Euclidean space that floats in equilibrium in any orientation (in water of density \(1\)) provided it is made of material of uniform density strictly between \(0\) and \(1\). More generally, the author proves that in each dimension \(d \geq 3\) there exists a set \(K \subset \mathbb{R}^d\) that is a strictly convex non-centrally-symmetric body of revolution and floats in equilibrium in every orientation at the level \(\frac{|K|}{2}\) where \(|K|\) denotes the \(d\)-dimensional volume. As a consequence, he obtains a result that in \(\mathbb{R}^3\) there exists a convex body \(M\) of density \(\frac12\) that witnesses the failure of the Ulam’s problem mentioned above. As was shown much earlier (by R. Schneider [Enseign. Math. (2) 16, 297–305 (1971; Zbl 0209.26502)] and K. J. Falconer [Am. Math. Mon. 90, 690–693 (1983; Zbl 0529.52001)]), no such \(M\) can be centrally symmetric. Other earlier results related with the topic of the paper are listed in the Introduction.

MSC:

52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
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