Abstract
Summary: We establish a lower bound theorem for the number of (k)-faces ((1\leqslant k\leqslant d-2)) in a (d)-dimensional polytope (P) (abbreviated as a (d)-polytope) with (2d+2) vertices, building on the known case for (k=1). We identify all minimisers for (d\leqslant 5). Two distinct lower bounds emerge, depending on the number of facets of (P). When (P) has precisely (d+2) facets, the lower bound is tight when (d) is odd. If (P) has at least (d+3) facets, the lower bound is always tight, and equality holds for some (1\leqslant k\leqslant d-2) only when (P) has precisely (d+3\ ) facets.
Moreover, for (1\leqslant k \leqslant \lceil d/3 \rceil-2), the minimisers among (d)-polytopes with (2d+2) vertices have precisely \ (d+3) facets, while for (\lfloor 0.4d\rfloor\leqslant k\leqslant d-1), the lower bound arises from (d)-polytopes with (d+2) facets.
Guillermo Pineda Villavicencio
Associate Professor in Computer Science and Mathematics
In May 2021, I became a Fellow (FHEA) of AdvanceHE, a member-led, sector-owned charity dedicated to enhancing higher education for staff, students, and society worldwide. Additionally, in 2016, I completed a Graduate Certificate in Education (Tertiary Teaching) at Federation University Australia (formerly University of Ballarat). I earned my Doctor of Philosophy in 2009 from Federation University Australia. In 2002, I graduated with a Bachelor’s degree in Computer Science with first-class honours from the University of Oriente, Santiago de Cuba.