Analytic Parametrization of Three-Dimensional Bodies of Constant Width

Bodies of constant width in arbitrary dimension

“…This is important in particular for variational studies, like the still open problem of finding the orbiform of minimal volume. We will use these characterizations in a forthcoming article on this problem [1].…”

“…The preceding characterizations are useful, but the diameter condition is difficult to handle in the variational context we consider in [1]. So let us give a slightly different characterization of bodies of constant width.…”

“…On the other hand there are some bodies of constant width whose group of symmetries is the same than the tetrahedron's. In order to construct one, it suffices to start from one of the Meissner's bodies K 1 , that have all the required symmetries except one. So let K 2 be the symmetrical of K 1 with respect to the missing plane of symmetry.…”

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

“…Moreover, the polar body of such a convex body is of constant width In fact, the question makes sense to ask for all 0 < w * < π. Thus, we have arrived at the following quite basic volume problem, whose Euclidean counterpart has been much better studied and is also better known (see for example [3]). …”

Analytical Parameterization of Rotors and Proof of a Goldberg Conjecture by Optimal Control Theory

“…The study of these problems in R 2 is useful for extensions in R 3 and in the domain of spectral analysis. For example, the problem of finding a constant width body of minimal volume in R 3 has recently been investigated (see [4], [22]). The optimization of eigenvalues with respect to the domain Ω is also an intense field of research (see [21] for an overview of many spectral problems involving convexity).…”