# On the complexity of random polytopes

## TL;DR

This paper provides an elementary overview of the complexity of random polytopes, focusing on their combinatorial and geometric properties, and suggesting that polytopes formed from random points on a sphere tend to have low complexity.

## Contribution

It offers a simplified, accessible introduction to the study of random polytopes, emphasizing their low complexity and avoiding complex technical methods.

## Key findings

- Random polytopes from sphere points have low complexity
- Elementary methods can analyze random polytopes effectively
- Insights into average-case complexity of polytopal algorithms

## Abstract

There are (at least) two reasons to study random polytopes. The first is to understand the combinatorics and geometry of random polytopes especially as compared to other classes of polytopes, and the second is to analyze average-case complexity for algorithms which take polytopal data as input. However, establishing results in either of these directions often requires quite technical methods. Here we seek to give an elementary introduction to random polytopes avoiding these technicalities. In particular we explore the general paradigm that polytopes obtained from the convex hull of random points on a sphere have low complexity.