Expected Number of Vertices of a Hypercube Slice

TL;DR

This paper determines that the expected number of vertices in a random k-dimensional slice of a hypercube is 2^k, regardless of the hypercube's dimension, under a specific distribution of slices.

Contribution

It establishes a dimension-independent expected vertex count for hypercube slices, revealing a surprising uniformity across dimensions.

Findings

Expected vertices of hypercube slices is 2^k

Result holds for a specific distribution of slices

Dimension of hypercube does not affect the expected vertex count

Abstract

Given a random k-dimensional cross-section of a hypercube, what is its expected number of vertices? We show that, for a suitable distribution of random slices, the answer is $2^k$, independent of the dimension of the hypercube.