We introduce a new method of proving lower bounds on the depth of algebraic $d$-degree decision trees and apply it to prove a lower bound $\Omega (\log N)$ for testing membership to an $n$-dimensional convex polyhedron having $N$ faces of all dimensions, provided that $N > (nd)^{\Omega (n)}$. This bound apparently does not follow from the methods developed by M. Ben-Or, A. Bj\"orner, L. Lovasz, and A. Yao [B. 83], [BLY 93], [Y 94] because topological invariants used in these methods become trivial for convex polyhedra.