Let $\hat{\mathbb{R}}^n$ be the one point compactification of $\mathbb{R}^n$ obtained by adding a point at infinity. We say that a subset $A\subseteq \hat{\mathbb{R}}^n$ is $\mathbf{u}$-convex if for every pair of points $\mathbf{z}_1, \mathbf{z}_2 \in A$, the arc of the unique circle through $\mathbf{u}, \mathbf{z}_1$ and $\mathbf{z}_2$, from $\mathbf{z}_1$ to $\mathbf{z}_2$ and not containing $\mathbf{u}$, is contained in $A$. In this case, we call $\mathbf{u}$ a pole of $A$. When the pole $\mathbf{u}$ approaches infinity, $\mathbf{u}$-convex sets become convex in the classical sense.

The notion of polar convexity in the complex plane has been used to analyze the behavior of critical points of polynomials. In this paper, we extend the notion to finite dimensional Euclidean spaces. The goal of this paper is to start building the theory of polar convexity and to show that the introduction of a pole creates a richer theory. For example, polar convexity enjoys a beautiful duality that does not exist in classical convexity. We formulate polar analogues of several classical results of the alternatives, such as Gordon and Farkas' lemmas. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles.

The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique -- two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible to use Lyapunov rank to identify cones in general. But perhaps that's not the full story. "Loewnerian" cones are defined as being isomorphic to the (symmetric) real positive-semidefinite cone and have applications in optimization if we can recognize them. Motivated by this, we ask a more limited question: among *symmetric* cones, are there any that can be uniquely identified (up to isomorphism) by their Lyapunov ranks?

In the theory of optimal transport, Rockafellar type constructions of antiderivatives are known to be solutions
of the dual problem. In multi-marginal optimal transport theory, it is still unknown how to construct
explicit c-splitting potentials in general. We present the problem and discuss cases where
solutions are known.

In various disciplines, data points live in nonpositively curved metric spaces - those in which triangles are thinner than their Euclidean analogues. In such spaces - also known as CAT(0) spaces, or as Hadamard spaces (when complete) - convex optimization techniques are particularly attractive, because, given any point in the space, the squared-distance to that point is a strongly convex function along geodesics. A common example of a nonpositively curved space is the space of positive-definite matrices with its standard affine-invariant metric - a Hadamard manifold. However, spaces that are not manifolds are also common, in models ranging from robotics to phylogenetics: an important example is the tree space of Billera-Holmes-Vogtmann (2001), which is a complex of Euclidean orthants. Nonpositively curved spaces support various natural notions of "averaging" for finite sets of points in the space. Unfortunately, among these, the convex hull operation is poorly understood (even for three points). Weighted means (or "barycenters"), on the other hand, are in principle computable, but although they have served as the motivation for extensive phylogenetic computation, existing general-purpose algorithms for weighted means outside the case of manifolds are very slow. In the light of these challenges, we discuss the tractability of weighted mean problems - both computing them and recognizing them - in some important nonpositively curved spaces.