Vladimir wins exactly when there is a pie with distance to the border not greater than 4. Indeed if there is such a pie, then Vladimir will move it to the border and then move it around the whole rim. Of course if there would be any chance to throw this pie from the board, Vladimir will use it. If he gets no such chance, then after mentioned moves all border is banned. But it means Vlad made at least 2n + 2m turns, when Vladimir only 2n + 2m - 1 as a maximum. A contradiction. Else, if there is no such pie, then during first 4 turns Vlad would ban sides adjacent to each corner of the board. After that, if some pie comes to the rim he would ban adjacent side (there is no more than one such side) ans so Vladimir will never get the pie.
So solution consists of only one distance check, but, as one may see, it is easy to make mistake.