Abstract
Summary: We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. 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 $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets. Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, 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.