Convexity of Sets Under Normal Distribution in the Structural Alloy Steel Standard

The paper is motivated by the structural alloy steel standard that has been used in China for a long period. This standard indicates the scope of several chemical elements in the steel and requests several mechanical properties for qualification. Fang and Wu (Acta Math Appl Sin 2:132–148, 1979) established the relationships between the percents of the controlled chemical elements and testing mechanical properties by a multivariate regression model, and proposed the algorithm for calculating qualification rate. Moreover, they proved the existence of the optimal chemical element combination. However, the uniqueness of the optimal solution for high dimensional case has been left. This open question is equivalent to showing the convexity of a type of probability sets under multivariate normal distribution. This paper proves that the open question is true.