## Computing the extreme distances between two convex polygons.(English)Zbl 0604.68079

A polygon in the plane is convex if it contains all line segments connecting any two of its points. Let P and Q denote two convex polygons. The computational complexity of finding the minimum and maximum distance possible between two points p in P and q in Q is studied. An algorithm is described that determines the minimum distance (together with points p and q that realize it) in O(log m$$+\log n)$$ time, where m and n denote the number of vertices of P and Q, respectively. This is optimal in the worst case. For computing the maximum distance, a lower bound $$\Omega (m+n)$$ is proved. This bound is also shown to be best possible by establishing an upper bound of $$O(m+n)$$.

### MSC:

 68R99 Discrete mathematics in relation to computer science 52A10 Convex sets in $$2$$ dimensions (including convex curves) 68Q25 Analysis of algorithms and problem complexity

### Keywords:

algorithm; minimum distance; maximum distance
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