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On a general Fano hypersurface in projective space, we determine forinfinitely many $k$ the minimal degree $e$ of a rational curve through ageneral collection of $k$ points. In the case of a hypersurface of index 1, ourresults hold for all $k\geq 1$. In an appendix, M.C. Chang proves anarithmetical result which implies that in the case of index $>1$, the densityof the set of curve degrees $e$ covered by our method is approximately$\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.

Balanced rational curves and minimal rational connectedness of Fano hypersurfaces