You can use an incremental approach.

In 3D, it works as follows:

• Assume that as some point you have constructed the (possibly open) polytope defined by the $k$ first plane.

• Consider the next plane and change the coordinate system so that the equation simplifies to $z=0$.

• For every edge of the polytope, check if the endpoints belong to the same half-plane defined by the next plane.

• If both have $z>0$, keep the edge; if both have $z<0$, discard the edge; otherwise, compute the intersection point of the edge with the plane and trim the edge. Finally, join the intersection points to form a convex section polygon.

• Repeat with the remaining planes.

This method cannot avoid the $O(n^3)$ behavior in the worst case (no method can) but if you are lucky it can keep the number of useless vertex computations to a low value. (At best, $O(nm)$ intersections for $n$ planes and $m$ vertices in the solution.)

You can use an incremental approach.

In 3D, it works as follows:

• Assume that as some point you have constructed the (possibly open) polytope defined by the $k$ first planes.

• Consider the next plane and change the coordinate system so that the equation simplifies to $z=0$.

• For every edge of the polytope, check if the endpoints belong to the same half-plane defined by the plane ($z$ sign).

• If both have $z>0$, keep the edge; if both have $z<0$, discard the edge; otherwise, compute the intersection point of the edge with the plane and trim the edge. Finally, join the intersection points to form a convex section polygon.

• Repeat with the remaining planes.

This method cannot avoid the $O(n^3)$ behavior in the worst case (no method can) but if you are lucky it can keep the number of useless vertex computations to a low value. (At best, $O(nm)$ intersections for $n$ planes and $m$ vertices in the solution.)

You can use an incremental approach.

In 3D, it works as follows:

• Assume that as some point you have constructed the (possibly open) polytope defined by the $k$ first plane.

• Consider the next plane and change the coordinate system so that the equation simplifies to $z=0$.

• For every edge of the polytope, check if the endpoints belong to the same half-plane defined by the next plane.

• If both have $z>0$, keep the edge; if both have $z<0$, discard the edge; otherwise, compute the intersection point of the edge with the plane and trim the edge. Finally, join the intersection points to form a convex section polygon.

• Repeat with the remaining planes.

This method cannot avoid the $O(n^d)$ behavior in the worst case (no method can) but if you are lucky it can keep the number of useless vertex computations to a low value. (At best, $O(nm)$ for $n$ planes and $m$ vertices in the solution.)

You can use an incremental approach.

In 3D, it works as follows:

• Assume that as some point you have constructed the (possibly open) polytope defined by the $k$ first plane.

• Consider the next plane and change the coordinate system so that the equation simplifies to $z=0$.

• For every edge of the polytope, check if the endpoints belong to the same half-plane defined by the next plane.

• If both have $z>0$, keep the edge; if both have $z<0$, discard the edge; otherwise, compute the intersection point of the edge with the plane and trim the edge. Finally, join the intersection points to form a convex section polygon.

• Repeat with the remaining planes.

This method cannot avoid the $O(n^3)$ behavior in the worst case (no method can) but if you are lucky it can keep the number of useless vertex computations to a low value. (At best, $O(nm)$ intersections for $n$ planes and $m$ vertices in the solution.)