Related papers: Depth with respect to a family of convex sets

The Tukey depth of a flat with respect to a point set is a concept that appears in many areas of discrete and computational geometry. In particular, the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges can all be…

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade…

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the…

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We…

A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it…

We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $\rho$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii…

Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted…

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence,…

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of B\'ar\'any, Katchalski, and Pach (1982). Improving on several recent works, we prove an…

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the…

Let $\mu$ be a Borel probability measure in $\mathbb R^d$. For a $k$-flat $\alpha$ consider the value $\inf \mu(H)$, where $H$ runs through all half-spaces containing $\alpha$. This infimum is called the half-space depth of $\alpha$. Bukh,…

We present a new fast approximate algorithm for Tukey (halfspace) depth level sets and its implementation-ABCDepth. Given a $d$-dimensional data set for any $d\geq 1$, the algorithm is based on a representation of level sets as…

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional…

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly…

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F $ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal…

Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a…

We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball etc. Our work relies…

Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets and let $\Delta(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The…

Let $K$ be a compact convex set in $\mathbb{R}^2$ and let $\mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3$ be finite families of translates of $K$ such that $A \cap B \neq \emptyset$ for every $A \in \mathcal{F}_i$ and $B \in \mathcal{F}_j$…

Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Sober\'{o}n proved a fractional version of Halman's theorem for axis-parallel boxes, showing…