<p><b>Abstract</b>—An algorithm for the linear interpolation of multi-input functions sampled on rectangular grids is presented. A geometric approach is adopted and the mathematics is thoroughly developed. We show that the algorithm is optimum. In fact, when the number <it>n</it> of inputs grows to infinity its computational requirement is <tmath>${\cal O}(n {\rm\ log}\ n),$</tmath> which is the same as the lower-bound on the cost of continuous linear interpolation procedures.</p>