Medians and means in Finsler geometry

International audienceWe investigate existence and uniqueness of p-means and the median of a probability measure on a Finsler manifold, in relation with the convexity of the support of the measure. We prove that the p-mean is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for larger than or equal to 2, a discretization of this path converges to the p-mean. This provides an algorithm for determining those Finsler center points

Geometric matrix midranges

We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the extension of the problem to $N > 2$ matrices. We compare matrix midrange statistics with the scalar and vector midrange problem and note the special significance of the matrix problem from a computational standpoint. We also study various aspects of geometric matrix midrange statistics from the viewpoint of linear algebra, differential geometry and convex optimization.ECH2020 EUROPEAN RESEARCH COUNCIL (ERC) (670645

Means in complete manifolds: uniqueness and approximation

International audienceLet $M$ be a complete Riemannian manifold, N\in \NN and $p\ge 1$. We prove that almost everywhere on $x=(x_1,\ldots,x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure \di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k} has a unique $p$-mean $e_p(x)$. As a consequence, if $X=(X_1,\ldots,X_N)$ is a $M^N$-valued random variable with absolutely continuous law, then almost surely \mu(X(\om)) has a unique $p$-mean. In particular if $(X_n)_{n\ge 1}$ is an independent sample of an absolutely continuous law in $M$, then the process e_{p,n}(\om)=e_p(X_1(\om),\ldots, X_n(\om)) is well-defined. Assume $M$ is compact and consider a probability measure $\nu$ in $M$. Using partial simulated annealing, we define a continuous semimartingale which converges to the set of minimizers of the integral of distance at power~$p$ with respect to $\nu$. When the set is a singleton, it converges to the $p$-mean

A stochastic algorithm finding generalized means on compact manifolds

International audienceA stochastic algorithm is proposed, finding the set of generalized means associated to a probability measure on a compact Riemannian manifold M and a continuous cost function on the product of M by itself. Generalized means include p-means for p>0, computed with any continuous distance function, not necessarily the Riemannian distance. They also include means for lengths computed from Finsler metrics, or for divergences. The algorithm is fed sequentially with independent random variables Y_n distributed according to the probability measure on the manifold and this is the only knowledge of this measure required. It evolves like a Brownian motion between the times it jumps in direction of the Y_n. Its principle is based on simulated annealing and homogenization, so that temperature and approximations schemes must be tuned up. The proof relies on the investigation of the evolution of a time-inhomogeneous L^2 functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock

Viterbo's conjecture for Lagrangian products in R4\mathbb{R}^4 and symplectomorphisms to the Euclidean ball

We analyze Viterbo's conjecture for the EHZ-capacity for convex Lagrangian products in $\mathbb{R}^4$. We use the generalized Minkowski billiard characterization of the EHZ-capacity in order to reprove that Viterbo's conjecture holds for the Lagrangian products (any triangle/parallelogram in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) and extend this fact to the Lagrangian products (any trapezoid in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$). Based on this analysis, we classify equality cases of Viterbo's conjecture and prove that most of them can be proven to be symplectomorphic to Euclidean balls. As a by-product, we prove sharp Minkowski billiard / worm problem inequalities. Finally, we discuss the Lagrangian products (any convex quadrilateral in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) for which we show that the truth of Viterbo's conjecture would follow from the positive solution of a challenging Euclidean packing problem.Comment: 57 pages, 31 figures, some corrections mad

Hyperbolic geometry in the work of Johann Heinrich Lambert

The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre-decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean axioms are kept except the parallel axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Th{\'e}ret, La th{\'e}orie des lignes parall{\`e}les de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014)