This study examines the compressive response of geometrically tailored pyramidal lattice structures composed of struts with I-shaped cross-sections. Geometrically tailored pyramidal lattices with micro-scale features are 3D printed using the Digital Light Processing (DLP) technique, and their effective elastic modulus, collapse strength and energy absorption capacity are experimentally evaluated under quasi-static compressive loading. Furthermore, detailed non-linear finite element (FE) calculations are performed to examine underlying collapse mechanisms and explore the vast design space offered by the proposed geometrical tailoring scheme. Both the experimental and numerical results show that the geometrically tailored lattice structures outperform conventional pyramidal lattices of equal weight in terms of elastic modulus (+24 %), collapse strength (+21 %) and energy absorption (+68 %). Notably, these strength improvements are attributed to lateral buckling that prompts the I-shaped struts to bend sideways during collapse. Specific cross-sectional designs demonstrate remarkable enhancements in strength and energy absorption, reaching up to 93 % and 161 %, respectively, differentiating them significantly from conventional designs.