Topological transversals to a family of convex sets

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 $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

Notes about the Caratheodory number

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 theorem and give a Tverberg type theorem for families of convex compacta

Computing a center-transversal line

A center-transversal line for two finite point sets in R 3 is a line with the property that any closed halfspace that contains it also contains at least one third of each point set. It is known that a center-transversal line always exists [14, 29], but the best known algorithm for finding such a line takes roughly n 12 time. We propose an algorithm that finds a center-transversal line in O(n 1+ε κ 2 (n)) worst-case time, for any ε&gt; 0, where κ(n) is the maximum complexity of a single level in an arrangement of n planes in R 3. With the current best upper bound κ(n) = O(n 5/2) of [26], the running time is O(n 6+ε), for any ε&gt; 0. We also show that the problem of deciding whether there is a center-transversal line parallel to a given direction u can be solved in O(n log n) expected time. Finally, we extend the concept of center-transversal line to that of bichromatic depth of lines in space, and give an algorithm that computes a deepest line exactly in time O(n 1+ε κ 2 (n)), and a linear-time approximation algorithm that computes, for any specified δ&gt; 0, a line whose depth is at least 1 − δ times the maximum depth