XV Spanish Meeting on Computational Geometry, June 26-28, 2013 Three location tapas calling for CG sauce Frank Plastria *1 1 MOSI - Vrije Universiteit Brussel,Pleinlaan 2, B 1050 Brussels, Belgium Abstract Based on some recent modelling considerations in lo- cation theory we call for study of three CG constructs of Voronoi type that seem not to have been studied much before. Introduction There is a strong interrelation and mutual ensemina- tion between (continuous) location theory and com- putational geometry. The first generates interest into distance optimization problems with geometrical interpretations, the second builds geometrical algo- rithms for efficient solution to the first’s problems. In this talk I will shortly present some recent work stemming from location theory calling for study of three novel (?) computational geometry constructs. 1 Mixed norm shortest paths The fact that the distance measure may be different from one region to another, contrary to the usual as- sumption that distance is measured by a single norm, has been acknowledged in only a few location stud- ies. Parlar [14] considers the plane divided by a linear boundary with at one side ℓ 2 and at the other side ℓ 1 . Brimberg et al [2] consider a bounded region with a different norm inside and outside, focusing in partic- ular on an axis-parallel rectangular city with ℓ 1 inside and ℓ 2 outside. Brimberg et al. [1] and Zaferanieh et al. [19] con- sider location in a space with two distinct ℓ p norms in complementary halfplanes. The way to calculate the distance in such a space was studied more in detail by Franco et al [7]. Fathali and Zaferanieh [5] extend this work to in- clude more general block norms. Fliege [6] consid- ers differentiably changing metrics similar to Riemann spaces. Unfortunately part of this work is wrong. All au- thors consider only two possibilities when calculating distances: when the two points lie in the same half- plane simply use the corresponding distance, and oth- * Email: Frank.Plastria@vub.ac.be erwise in two steps via the best possible point on the separating line. Although this is true in some partic- ular cases, e.g. the axis-parallel rectangular city case evoked before (but without inflation factors), Parlar already observed that in general when calculating dis- tance in this naive way distance to a fixed point is not continuous everywhere. Worse: triangle inequal- ity may be violated. This clearly shows that such distance calculation cannot be correct, but none of the authors try to resolve this discrepancy. What should rather be done is to consider shortest path distance in the space, similar to what is done in the so-called weighted (euclidean) region problem well known in CG since the original paper of Mitchell and Papadimitriou [10] (which depicts a clear coun- terexample to the naive distance above): the length of the shortest possible piecewise linear path using the adequate measure (speed in that paper) in each piece. For two halfspaces with arbitrary distinct norms or gauges we studied more in detail the optimality condi- tions when crossing the boundary, generalizing Snell’s law in optics. This has nice geometric interpretation and leads to a geometrical view on deriving such dis- tances [16]. What turns out to be crucial is the comparison be- tween the two norms along (the direction of) the sep- arating line. As long as these are equal things remain relatively simple and the naive assumption about dis- tances is correct. However, when they differ things change. One region is then ‘slower’ than the other (as measured along the boundary line). And in such a case some paths connecting two points within the slowest region will consist of three pieces. They ‘hitch a ride’ along the quicker boundary. Thereby continuity of the distance is again guaran- teed, but now convexity is partially lost, as illustrated by the balls in the figure below. At right of the vertical separation line we have the slower region in gray with distance measure 4ℓ 1 , at left the faster region with norm ℓ 2 . The figure shows balls of increasing mixed- norm radius centered at the dot. For very small radius we have the diamond-shaped ℓ 1 ball. As soon as the radius allows to reach the boundary a part of circular shape arises at left due to two piece shortest paths, which spills over with a linearly moving front at the right corresponding to three piece paths. The white line shows the set of meeting points where one-piece 31