Notice that the Manhattan distance between two points <code>[x_i, y_i]</code> and <code>[x_j, y_j] is <code> max({x_i - x_j + y_i - y_j, x_i - x_j - y_i + y_j, - x_i + x_j + y_i - y_j, - x_i + x_j - y_i + y_j})</code></code>.

If you replace points as <code>[x_i - y_i, x_i + y_i]</code> then the Manhattan distance is <code>max(max(x_i) - min(x_i), max(y_i) - min(y_i))</code> over all <code>i</code>.

After those observations, the problem just becomes a simulation. Create multiset of points <code>[x_i - y_i, x_i + y_i]</code>, you can iterate on a point you might remove and get the maximum Manhattan distance over all other points.