The paper by J. Kiefer and J. Wolfowitz explores the equivalence of two extremum problems in the context of probability measures and linear functions. The authors consider a set of linearly independent real functions \( f_1, \ldots, f_k \) defined on a compact space \( X \), and a class of probability measures \( \mathcal{C} \) that includes measures with finite support. The first extremum problem involves maximizing the determinant of the matrix \( M(\xi) \), which is defined as the Gram matrix of the functions \( f_i \) with respect to the measure \( \xi \). The second problem involves minimizing the maximum value of \( d(x; \xi) \), where \( d(x; \xi) \) is a function related to the matrix \( M(\xi) \). The main result of the paper is that the two extremum problems are equivalent, meaning that a measure \( \xi \) that maximizes the determinant of \( M(\xi) \) is also the one that minimizes \( \max_x d(x; \xi) \). This equivalence is established through a detailed proof that involves analyzing the properties of the matrix \( M(\xi) \) and the function \( d(x; \xi) \). The authors also provide a corollary that extends the result to the case where the functions \( f_i \) are continuous and defined on a compact space. This corollary states that there exists a probability measure \( \xi \) and a linear transformation such that the transformed functions are orthonormal with respect to \( \xi \) and the maximum value of the sum of their squares is \( k \). Finally, the paper discusses the statistical applications of these results, particularly in the design of regression experiments. The authors show that the two optimality criteria—minimizing the determinant of the covariance matrix and minimizing the maximum variance of the best linear estimator—are equivalent. This equivalence is useful for practical design optimization, as it allows for the use of one criterion to find an initial solution and then verify it using the other criterion.The paper by J. Kiefer and J. Wolfowitz explores the equivalence of two extremum problems in the context of probability measures and linear functions. The authors consider a set of linearly independent real functions \( f_1, \ldots, f_k \) defined on a compact space \( X \), and a class of probability measures \( \mathcal{C} \) that includes measures with finite support. The first extremum problem involves maximizing the determinant of the matrix \( M(\xi) \), which is defined as the Gram matrix of the functions \( f_i \) with respect to the measure \( \xi \). The second problem involves minimizing the maximum value of \( d(x; \xi) \), where \( d(x; \xi) \) is a function related to the matrix \( M(\xi) \). The main result of the paper is that the two extremum problems are equivalent, meaning that a measure \( \xi \) that maximizes the determinant of \( M(\xi) \) is also the one that minimizes \( \max_x d(x; \xi) \). This equivalence is established through a detailed proof that involves analyzing the properties of the matrix \( M(\xi) \) and the function \( d(x; \xi) \). The authors also provide a corollary that extends the result to the case where the functions \( f_i \) are continuous and defined on a compact space. This corollary states that there exists a probability measure \( \xi \) and a linear transformation such that the transformed functions are orthonormal with respect to \( \xi \) and the maximum value of the sum of their squares is \( k \). Finally, the paper discusses the statistical applications of these results, particularly in the design of regression experiments. The authors show that the two optimality criteria—minimizing the determinant of the covariance matrix and minimizing the maximum variance of the best linear estimator—are equivalent. This equivalence is useful for practical design optimization, as it allows for the use of one criterion to find an initial solution and then verify it using the other criterion.