In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering
$$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$
This resulted in
$$ p_k=-\frac{\nabla f(x_k)}{||\nabla f(x_k)||_2} $$ I am wondering if there is any significance of the Euclidean norm being used. I.E. would a different norm, such as the 1-norm or maximum(infinity)-norm result in a different steepest descent direction vector?