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arXiv · 2510.16672

A four-dimensional body of constant width

Marcela G. Mercado-Flores·Miguel Raggi·Edgardo Roldán-Pensado
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Abstract

The study of bodies of constant width is a classical subject in convex geometry, with the 3-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $\mathbb R^4$, addressing the challenge of constructing such bodies in higher dimensions. Our method produces a natural analogue of the second Meissner body, by modifying a 4-dimensional Reuleaux simplex. The resulting body possesses tetrahedral symmetry and has a boundary composed of both smooth surfaces and a non-smooth subset of the Reuleaux 4-simplex. Furthermore, we analyze the orthogonal projection of this body onto the 3-dimensional hyperplane of its base. This "shadow" is a 3-dimensional body of constant width with tetrahedral symmetry. It has six elliptical edges and its volume is only slightly larger than that of the Meissner bodies. This body was recently constructed as a projection of a different 4-dimensional body, however the construction presented here is new and gives additional properties.

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Marcela G. Mercado-Flores, Miguel Raggi, Edgardo Roldán-Pensado. 2025-10-19. A four-dimensional body of constant width. https://arxiv.org/abs/2510.16672

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