Beniamin BOGOSEL

Meissner polyhedra are constant-width bodies obtained from extremal finite sets of unit diameter. Such a generating set may contain dangling points, namely points having exactly two diametric neighbors. This article studies whether these points can play an essential role in surface-area minimization. Given an extremal set we show that the smallest surface area among the Meissner polyhedra based on it cannot increase by deleting a dangling point. Adding a dangling point cannot decrease the smallest achievable surface area. This reduces the search for area-minimizing Meissner polyhedra to generating sets without dangling points.

In this article it is shown that the equilateral triangle maximizes the Cheeger constant and minimizes the torsional rigidity among shapes having a fixed minimal width. The proof techniques use direct comparisons with simpler shapes, consisting of disks with three disjoint caps. Comparison results for harmonic functions help establish that in non-equilateral configurations the shape derivative has an appropriate sign, contradicting optimality.

The main result of the paper shows that the regular n-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among n-gons having fixed area for n∈{5,6}. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb{R}^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for n∈{5,6}. All computations, including the finite element computations, are realized using interval arithmetic.

The goal of this paper is to provide a qualitative analysis of the optimisation of space-time periodic principal eigenvalues. Namely, considering a fixed time horizon $T$ and the $d$-dimensional torus, let, for any $m\in L^\infty((0:T)\times\Omega)$, $\lambda(m)$ be the principal eigenvalue of the operator $\partial_t-\Delta-m$ endowed with (time-space) periodic boundary conditions. The main question we set out to answer is the following: how to choose $c$ so as to minimise $\lambda(m)$? This question stems from population dynamics. We prove that in several cases it is always beneficial to rearrange $m$ with respect to time in a symmetric way, which is the first comparison result for the rearrangement in time of parabolic equations. Our results are illustrated by several numerical simulations.

The first main result presented in the paper shows that the perimeters of inner parallel sets of planar shapes having a given constant width are minimal for the Reuleaux triangles. This implies that the areas of inner parallel sets and, consequently, the inverse of the Cheeger constant are also minimal for the Reuleaux triangles. Proofs use elementary geometry arguments and are based on direct comparisons between general constant width shapes and the Reuleaux triangle.