By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So I ask, is the maximum attained only at finitely many point for order above $3$, or perhaps, for orders not equal to $3$ mod 4? I suspect this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So, I ask, is the maximum attained only at finitely many points for order above $3$, or perhaps for orders not equal to $3$ mod 4? I suspect this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at some {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So I ask, is the maximum attained only at finitely many point for order above $3$, or perhaps, for orders not equal to $3$ mod 4? I suppose this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So I ask, is the maximum attained only at finitely many point for order above $3$, or perhaps, for orders not equal to $3$ mod 4? I suspect this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?

By multilinearity, the maximum of the determinant of matrices with entries in the interval [-1, 1] is attained at some {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So I ask, is the maximum attained only at finitely many point for order above $3$, or perhaps, for orders not equal to $3$ mod 4? I suppose this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at some {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many points on matrices of order $3$ \begin{bmatrix} 1 & 1 & 1\\ -1 & 1 & 1\\ t & 1 & -1 \end{bmatrix} $-1 \leq t \leq 1$. However, I verified that there are only finitely many maximum points for orders $4, 5, 6$. So I ask, is the maximum attained only at finitely many point for order above $3$, or perhaps, for orders not equal to $3$ mod 4? I suppose this is known, as there has been a lot of work on Hadamard Maximum determinant problem, but I can't find such statement so far. If this is not known, can we at least prove that the dimension of the space of maximum points is less than the order?