On the max min of the algebraic degree and the nonlinearity of a Boolean function on an affine subspace

We investigate the max min of the algebraic degree and the nonlinearity of a Boolean function in $n$ variables when restricted to a $k$-dimensional affine subspace of ${\mathbb{F}}_2^n$. Previous authors have focused on the cases when the max min of the algebraic degree is 0 or 1. Upper bounds, lower bounds and a conjecture on the exact value in special cases are presented.