  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 $\textbf{F}_2^n$. 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 is presented.
