The paper presents a novel heuristic procedure (further called the AH Method) to investigate function shape in the direct vicinity of the found optimum solution. The survey is conducted using only the space sampling collected during the optimization process with an evolutionary algorithm. For this purpose the finite model of point-set is considered. The statistical analysis of the sampling quality based upon the coverage of the points in question over the entire attraction region is exploited. The tolerance boundaries of the parameters are determined for the user-specified increase of the objective function value above the found minimum. The presented test-case data prove that the proposed approach is comparable to other optimum neighborhood examination algorithms. Also, the AH Method requires noticeably shorter computational time than its counterparts. This is achieved by a repeated, second use of points from optimization without additional objective function calls, as well as significant repository size reduction during preprocessing.