We demonstrate the utility of imposing epicentral distance constraints upon the delay time extremal inverse problem. The inference of delay times from a seismic profile is straightforward and stable since the association of these times with specific ray parameters is unnecessary. The estimation of distance data at specific ray parameters is a much noisier and unstable process, but can provide coarse constraints upon the inverse problem. These distance data act as linear smoothing constraints upon the delay time inverse problem. Furthermore, if a triplication exists in the observed travel-time data, the requirement of such a triplication can be imposed upon the extremal inverse problem using linear programming. Travel-time data from Eastern Iceland and the Pacific Basin are used to demonstrate the value of these constraints.