Related papers: Caratheodory's Theorem in Depth

We present a selection theorem for domains in $\mathbb{C}^n$, $n\ge 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carath\'eodory. Not only is…

Tukey depth, aka halfspace depth, has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample,…

Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in general position in $d$-space. It is shown that there exist a point $O$ and suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that $|Q_i|\geq…

We prove two colorful Carath\'eodory theorems for strongly convex hulls, generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre B\'ar\'any, the non-colorful Carath\'eodory theorem for strongly convex hulls by the…

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$ H(X_1+\cdots+X_{n+1}) \geq…

Suppose $d+1$ absolutely continuous probability measures $m_0, \ldots, m_d$ on $\mathbb{R}^d$ are given. In this paper, we prove that there exists a point of $\mathbb{R}^d$ that belongs to the convex hull of $d+1$ points $v_0, \ldots, v_d$…

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory…

The depth of a simple algebraic extension $(L/K,v)$ of valued fields is the minimal length of the Mac Lane-Vaqui\'e chains of the valuations on $K[x]$ determined by the choice of different generators of the extension. In a previous paper,…

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here.…

Let $C_1,\dots,C_{d+1}\subset \mathbb{R}^d$ be $d+1$ point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence $p_1, \dots, p_{d+1}$ with $p_i \in C_i$, for $i = 1, \dots, d+1$, a…

Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ any field. A first known lower bound of LS-category $cat(X)$ is the Toomer invariant $e_{\mathbb{K}} (X)$ (\cite{Too}). In $1980$'s F\'elix et al. introduced the…

Erd\"{o}s proved that for every infinite $X \subseteq \mathbb{R}^d$ there is $Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from $Y$ have distinct distances, and he gave partial results for general $a$-ary volume. In this…

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and…

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical…

The point selection theorem says that the convex hull of any finite point set contains a point that lies in a positive proportion of the simplices determined by that set. This paper proves several new volumetric versions of this theorem…

Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $\tau(C_1)\cdot \tau(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $\tau(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

Let $\# K$ be a number of integer lattice points contained in a set $K$. In this paper we prove that for each $d\in {\mathbb N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset…

It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i \in [0,q)$ satisfies the equality $\sum_{i=1}^\infty c_iq^{-i}=1$. The set of such "univoque numbers" has…

We prove that if $X$ is a topological space that admits Debreu's classical utility theorem (eg.\ $X$ is separable and connected, second countable, etc.), then order relations on $X$ satisfying milder completeness conditions can be…