Much of Welzl's research has been in
computational geometry. With
David Haussler, he showed that machinery from
computational learning theory including
ε-nets and
VC dimension could be useful in geometric problems such as the development of space-efficient
range searching data structures. He devised
linear time randomized algorithms for the
smallest circle problem and for low-dimensional
linear programming, and developed the combinatorial framework of
LP-type problems that generalizes both of these problems. Other highly cited research publications by Welzl and his co-authors describe algorithms for constructing
visibility graphs and using them to find shortest paths among obstacles in the plane, test whether two point sets can be mapped to each other by a combination of a geometric transformation and a small perturbation, and pioneer the use of
space-filling curves for range query data structures. ==Awards and honors==