# Monotonicity of expected $f$-vectors for projections of regular polytopes

## TL;DR

This paper proves that the expected number of faces of projected regular polytopes increases with dimension, and extends this monotonicity to Gaussian polytopes and zonotopes, revealing fundamental geometric properties.

## Contribution

It establishes the monotonicity of expected face counts for projections of regular polytopes and related Gaussian structures, a novel insight in geometric probability.

## Key findings

- Expected face counts increase with polytope dimension.
- Monotonicity holds for Gaussian polytopes and zonotopes.
- Results provide new understanding of geometric properties of random polytopes.

## Abstract

Let $P_n$ be an $n$-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project $P_n$ onto a random, uniformly distributed linear subspace of dimension $d\geq 2$. We prove that the expected number of $k$-dimensional faces of the resulting random polytope is an increasing function of $n$. As a corollary, we show that the expected number of $k$-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope.