The work of this research covers the Maximum Subarray Problem (MSP) from a new perspective. Research done previously and current methods of finding MSP include using the rectangular shape for finding the maximum sum or gain. The rectangular shape region used previously is not flexible enough to cover various data distributions. This research suggested using the convex shape, which is expected to have optimised and efficient results. In this research, the following findings are achieved: the first achievement is presenting an efficient algorithm, which determines the boundaries of the convex shape while having the same time complexity as that for other existing algorithms (the prefix sum was used to speed up the convex shape algorithm in finding the maximum sum). The second achievement is generalizing the algorithm to find up to the Kth maximum sum. Finding the Kth maximum convex sum was shown to be useful in many applications, such as data mining, and potentially locating brain tumours accurately.