J. Hinman

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math.CO2025

Fat Shellable Spheres

Joshua Hinman

The fatness of a 4-polytope or 3-sphere is defined as $(f_{1} + f_{2} - 20) / (f_{0} + f_{3} - 10)$ . We construct arbitrarily fat, strongly regular CW 3-spheres that are both shellable and dual shel…

math.CO2025

Reconstructing Polytopes and Pseudomanifolds

Joshua Hinman

We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each $d \geq 3$ , we show that not every $d$ -polytope is determ…

math.CO2024

Face Numbers of Shellable CW Balls and Spheres

Joshua Hinman

Let $X$ be the boundary complex of a $(d + 1)$ -polytope, and let $ρ (d + 1, k) = \frac{1}{2} [(d - k ⌈( d + 1 ) /2 ⌉) + (d - k ⌊( d + 1 ) /2 ⌋)]$ . Rece…

math.CO2024

Lower Bounds on Face Numbers of Polytopes with $m$ Facets

Joshua Hinman

Let $P$ be a convex $d$ -polytope and $0 \leq k \leq d - 1$ . In 2023, this author proved the following inequalities, resolving a question of Bárány: \[ \frac{f_k(P)}{f_0(P)} \geq \fra…

Fat Shellable Spheres

Reconstructing Polytopes and Pseudomanifolds

Face Numbers of Shellable CW Balls and Spheres

Lower Bounds on Face Numbers of Polytopes with m Facets

Lower Bounds on Face Numbers of Polytopes with $m$ Facets