We study the geometry of the space of measures of a compact ultrametric space $X$, endowed with the $L^{p}$ Wasserstein distance from optimal transportation. We show that the power $p$ of this distance makes this Wasserstein space affinely isometric to a convex subset of $\ell ^{1}$. As a consequence, it is connected by $1/p$-Hölder arcs, but any ${\it\alpha}$-Hölder arc with ${\it\alpha}>1/p$ must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when $X$ is ultrametric; however, thanks to the Mendel–Naor ultrametric skeleton it has consequences even when $X$ is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate, which needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of regular ultrametric spaces, i.e. spaces of the form $\{1,\dots ,k\}^{\mathbb{N}}$ with a natural ultrametric. We are also led to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space (). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a positive quadratic expression in the support function h and its derivative, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body K1 of finite perimeter, the set in this class that is farthest away in the sense of the L2 distance is always a line segment. The same property is proved for the Hausdorff distance.

In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

We prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

We characterize least-perimeter enclosures of prescribed area on some piecewise smooth manifolds, including certain polyhedra, double spherical caps, and cylindrical cans.