There is a deep connection between the mathematical notion of convexity, and practical tractability of solving an optimization problem. Thanks to new primal-dual interior point methods developed and refined over the past two decades, many classes of convex optimization problem can now be solved in polynomial time on a laptop computer. Our research leverages one such problem class, the geometric program (GP), to reliably and efficiently explore conceptual aircraft design spaces. This approach provides a rich formalism for casting engineering design problems, including guarantees of global optimality, sensitivity analysis via Lagrange duality, and fitting of GP-compatible surrogate models to data. Limitations and extensions to non-convex design spaces will also be discussed.