We describe a new method of proving lower bounds on the depth of algebraic decision trees and apply it to prove a lower bound $\Omega (\log N)$ for testing membership to a convex polyhedron having $N$ facets of all dimensions. This bound apparently does not follow from the methods developed by M. Ben-Or, A. Björner, L. Lovasz, A. Yao ([B 83], [BLY 92]) because the topological invariants used in these methods become trivial for the convex polyhedra.