All Questions

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth? A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...

Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is ...

Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices. The function $f(\cdot )$ is concave ...

Given $\mathbb{S}^n_+:=\{x\in \mathbb{R}^{n+1}: |x|=1,x_{n+1}>0\}$ be the open domain in $\mathbb{S}^n$, or be viewed as the geodesic ball centered at the pole with radius $\frac{\pi}{2}$ in $\...

Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...