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The Minkowski asymmetry of a convex body K in R^d, i.e., the smallest dilation factor λ for which K is contained in a translate of λ(-K), is known to be at most d. The maximizers of the Minkowski symmetry are precisely the simplices, in particular those which are mirror symmetric with respect to some hyperplane. In order to quantify how mirror symmetric a given convex body K in R^d is, we may study the smallest dilation factor λ for which K is contained in a translate of some λ A_L(K) where A_L is the reflection about some j-dimensional linear subspace L of R^d. The Minkowski asymmetry is the j=0 case, and in this talk we focus on the case where d=2, j=1, and K is a triangle. We present some numerical evidence and discuss our analytical findings.

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We consider the problem of making nonparametric inference in multi-dimensional diffusion models from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity (in divergence form models), we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors. Interestingly, the optimisation schemes provided satisfactory numerical recovery while exhibiting rapid convergence towards stationary points despite the problem nonlinearity; thus our approach may lead to significant computational speed-ups.