This is the first volume of the proceedings of the Committee on the Undergraduate Program in Mathematics (CUPM) Geometry Conference, held at Santa Barbara in June, 1967. The purpose of the conference was to consider the status of geometry in colleges at the undergraduate level. The conference, attended by undergraduate mathematics teachers, involved lectures on various aspects of geometry, analysis of material presented, and an examination of the relevance of geometry to the undergraduate curriculum. In Part I of the proceedings are contained (1) an introduction by Walter Prenowitz and (2) the lectures on Convex Sets and the Combinatorial Theory of Convex Polytopes and Applications of Geometry, by Branko Grunbaum and Victor Klee. (RP)