I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the s …

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which …

Let $C$ and $D$ be two convex sets. And suppose $C\cap D\neq \emptyset$. Let $x^*$ is the solution to the optimization problem: $$\min_{x\in C} \max_{y \in D} |x-y|^2$$ Is it tru …

Assume we have a centrally symmetric convex set $K \subset \mathbb{R}^n$ such that Vol(K)=1. In addition, assume that for every direction $u$ we know that $Vol(K \Delta R_u(K)) &lt …

Hi, This is a problem that has being bothering me the last few days. Assume a convex function $f(x): {\mathbb R}^n \rightarrow {\mathbb R}$ with a unique minimizer $x^{\star}$ …

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{ …

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when c …

Background: Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+} …

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots …

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that …

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach …

Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose ${x_1,\ldots,x_n} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots, …

Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$? I have just fi …

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the follo …

Hi, I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the …