In order to realize the optimal luminance-equalized color conversion for multi-primary color display, a convex hull method to compute the display gamut volume is proposed, and a color conversion algorithm by nonlinear optimization of a luminance equalization merit function is proposed. First, the color coordinates are uniformly sampled in XYZ color space, and the color coordinates located in the device gamut volume are judged by the convex hull algorithm. Then, the luminance-equalized drive values corresponding to the color coordinates in device gamut volume are calculated to establish the mapping from XYZ color space to device color space. Color gamut computing and luminance-equalized color conversion are realized with a six-primary LED display. The results show that the proposed algorithm can achieve color conversion with a luminance variation coefficient of 0.1685 averagely.