Beniamin BOGOSEL

We study optimization problems for separable functionals of the Euclidean or spherical lengths of dual edge pairs in finite extremal unit-diameter configurations in three dimensions. For a fixed diameter graph, these problems lead to nonconvex constrained optimization of the vertex coordinates. We first prove that a convergent sequence of extremal configurations retains an extremal geometric core after coincident points are merged and vertices incident to at most one diameter are removed. A spherical Crofton argument then gives a sharp lower bound for additive concave functionals, attained by the regular tetrahedron. We also analyze the effect of inserting or deleting dangling vertices. For the sum of products of spherical dual-edge lengths, we obtain an exact supremal reformulation of the three-dimensional Blaschke–Lebesgue area problem. The numerical study uses all 10,644 available extremal configurations with at most 16 vertices. We combine direct evaluation with gradient-based local optimization on each fixed graph and reconstruct the intrinsic diameter graph and its dual pairs after vertex collisions. For every supplied graph, the selected verified endpoint contains a regular tetrahedron. These computations do not certify the global maximum for any fixed graph, but they motivate structural conjectures connecting tetrahedral containment with the Blaschke–Lebesgue problem.

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.