Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of $3$-polytopes with edges tangent to the unit sphere.
Combinatorics Algebraic Geometry 52B10 (Primary) 14P10 (Secondary)
A longstanding goal of biology is to identify the key genes and species that critically impact evolution, ecology, and health.
We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega(2^r)$.
Optimization and Control Combinatorics 90C51, 14T05