In this paper, the author establishes new properties of convex multivariate polynomials, using convex analysis. The author shows that for a convex polynomial \(f\) which is not constant on any affine subspace, if the lower level set of \(f\) (i.e., the set where \(f\) is nonpositive) is unbounded, then \(f\) can be represented as a sum of a convex polynomial in fewer variables and a linear form with negative coefficients. In Theorem 4.4, the author proves that for an \(m\)th-order convex polynomial \(f\), if there is a point \(x\) such that \(f(x)<0\), then \(f\) has a linear error bound. Otherwise, \(f\) has a local error bound of order \(1/m\). In Section 4, various types of error bounds for unconstrained and polyhedral-constrained convex polynomials are established.