Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic

Shortest secure path in a Voronoi Diagram

We investigate the problem of computing the shortest secure path in a Voronoi diagram. Here, a path is secure if it is a sequence of touching Voronoi cells, where each Voronoi cell in the path has a uniform cost of being secured. Importantly, we allow inserting new sites, which in some cases leads to significantly shorter paths. We present an O(nlogn) time algorithm for solving this problem in the plane, which uses a dynamic additive weighted Voronoi diagram to compute this path. The algorithm is an interesting combination of the continuous and discrete Dijkstra algorithms. We also implemented the algorithm using CGAL

Do Heads Roll? An Empirical Analysis of CEO Turnover and Pay When the Corporation is Federally Prosecuted

Does the criminal prosecution of a corporation affect the CEO? Or do criminal actions directed at the organization itself pose few consequences for the individuals at the top, and the CEO in particular? While CEOs are rarely themselves prosecuted, organizations could discipline CEOs through paycuts or outright replacing the CEO in response to a criminal prosecution. We sought to examine whether and how that occurs. We focus our analysis on a dataset of public companies that settled criminal cases brought by federal prosecutors from 2000-2014. We compared those companies to the larger set of companies in the Execucomp database of S&P 1500 firms, focusing on CEO compensation and turnover during the same time period. We examined the time period before and after prosecution, and the year that the company resolved the criminal charges against the company. We found that in the year that the company settled its prosecution, through a guilty plea or a deferred or non-prosecution agreement, there was a significantly higher level of CEO turnover. However, we do not find evidence of CEO pay cut. Second, for the prosecuted firms that did not have CEO turnover after prosecution, there is no evidence of a reduction in compensation. Indeed, we observed a spike in CEO bonuses in the year of prosecution—confirming concerns expressed by judges, prosecutors, lawmakers, and academics that corporate prosecutions do not sufficiently impact high-level decision-makers like CEOs. For the prosecuted firms that did have CEO turnover after prosecution, there is some evidence of a pay cut, both to salary and bonus, prior to the replacement of the CEO. These results raise larger questions whether federal prosecutors targeting the most serious corporate crimes sufficiently incentivize accountability at the top