This paper delves deeply into the convexity and smoothness characteristics of L p spaces. Through rigorous mathematical derivations, it comprehensively expounds on the convexity determination methods under different situations, including the proof of uniform convexity and the analysis of special cases when p=1 and p= ∞ . Meanwhile, it reveals the relationship between convexity and integrability. For smoothness, the Gateaux differentiability and Frechet differentiability are studied in detail, and the intrinsic relationship between convexity and smoothness is thoroughly explored. These research results are helpful for deepening the understanding of the geometric structure of L p spaces and are of great significance to the theory of functional analysis and its related application fields.