Abstract
In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.
Highlights
Abstract. — In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius
Inradius ρ run in the closed interval [1−1/ 3, 1/2]: the left endpoint, 1−1/ 3 ∼ 0.422, is the inradius of the Reuleaux triangle, which is well known to be the minimizer of the inradius among bodies of fixed constant width; as for the right endpoint, it is an easy consequence of (√1.1)
For the extremal values of r, the minimizer is known: on one hand, for r = 1 − 1/ 3, the optimal shape is the Reuleaux triangle, from Blaschke-Lebesgue theorem; on the other hand, for r = 1/2, it is clearly the disk of radius 1/2 which is the only set in the corresponding annulus
Summary
Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org. We should for example approximate any body of constant width by a sequence of Reuleaux polygons with increasing area lying in the same minimal annulus To clarify this result, let us show some picture. The fundamental proposition in our approach is the following, in which we give a characterization of rigid shapes It shows that we can restrict the study of optimal shapes to Reuleaux polygons having only extremal arcs and clusters. Let us prove that if a set has two consecutive vertexes, say Pk and Pk+1 lying in the interior of the annulus, it cannot be rigid In such a case the two arcs Γk and Γk+1 are not tangent and the Blaschke deformation described in Definition 2.4 is admissible in both senses (ε > 0 or ε < 0) without violating the annulus constraint.
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