# On the Complexity of Recognizing Nerves of Convex Sets

## TL;DR

This paper investigates the computational complexity of recognizing nerves of convex sets in Euclidean space, unifies previous results, and determines the complexity for various parameters, including proving ER-completeness in certain cases.

## Contribution

It unifies prior work under a common framework and settles the complexity of recognizing nerves of convex sets for all dimensions, including proving ER-completeness for specific cases.

## Key findings

- Recognizing nerves of convex sets is ER-complete for certain parameters.
- The complexity status of recognizing intersection graphs of convex sets is fully determined.
- The paper identifies trivial cases and generalizes previous results.

## Abstract

We study the problem of recognizing whether a given abstract simplicial complex $K$ is the $k$-skeleton of the nerve of $j$-dimensional convex sets in $\mathbb{R}^d$. We denote this problem by $R(k,j,d)$. As a main contribution, we unify the results of many previous works under this framework and show that many of these works in fact imply stronger results than explicitly stated. This allows us to settle the complexity status of $R(1,j,d)$, which is equivalent to the problem of recognizing intersection graphs of $j$-dimensional convex sets in $\mathbb{R}^d$, for any $j$ and $d$. Furthermore, we point out some trivial cases of $R(k,j,d)$, and demonstrate that $R(k,j,d)$ is ER-complete for $j\in\{d-1,d\}$ and $k\geq d$.