Characterizing the spatial structures of competing football teams.

Progress in football analytics is hampered by two main reasons: the unavailability of detailed datasets; and a dearth of meaningful metrics that would aid in gaining insight into the collective action of teams. We analyzed a dataset consisting of the full trajectories of all players, from twelve complete matches from three major international tournaments. We studied the teams' spatial structure, a topic that has been, to date, relatively unexplored. What has typically been analyzed is the teams' convex hull which gives a measure for the overall spread of the team in space, but which ignores the contributions of the players that are within it. Here we computed, for the first time, also the convex hull of those interior players. We then calculated the ratio of the area of the inner convex hull over the area of the outer convex hull, which gives us a single number that encapsulates the geometry of a team for a given point in time. We find a relatively general pattern-the ratio of the areas of these two convex layers almost always dies out at ~ 0.5 (i.e., there are almost no cases where the area of the inner layer exceeds 50% of the area of the outer layer). This result proves to be robust: for instance, it appears in both attacking and defending layouts (which are typically associated with very different team spatial structures). While it remains unclear whether a layer ratio value outside the observed range has practical implications, our findings provide a foundation for future research into the spatial geometry of team sports and its potential strategic relevance.