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We show how circumradius and asymptotic behavior of curves in CAT(0) andCAT(K) spaces(K>0) are controlled by growth rates of total curvature. We applyour results to pursuit and evasion games of capture type with simple pursuitmotion, generalizing results that are known for convex Euclidean domains, andobtaining results that are new for convex Euclidean domains and hold on playingfields vastly more general than these.

Total curvature and simple pursuit on domains of curvature bounded above