# An isoperimetric problem on the hypercube

Fix integer $n\ge 1$, and let $E=\{e_1,...,e_n\}$ denote the standard basis of the vector space ${\mathbb F}_2^n$. Thus, for a set $A\subset{\mathbb F}_2^n$, the sumset $A+E:=\{a+e\colon a\in A,\ e\in E\}$ consists of all those elements of ${\mathbb F}_2^n$ which are at Hamming distance $1$ from an element of $A$. Now I wonder,

How small can $|A+E|$ be in terms of $|A|$? How to choose a set $A$ of prescribed size to minimize the size of $A+E$?

I would expect the answer to be well-known -- any reference?

Somewhat closer to what I actually need is the situation where $A$ consists of even vectors only; that is, of vectors orthogonal to $e_1+\dotsb+e_n$. How to choose $A$ (of prescribed size) under this additional constrain to minimize $|A+E|$?

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If $E$ consists not only of unit vectors, but also of the zero vector, then according to Alon & Spencer Probabilistic method'' chapter 7, the sharp isoperimetric inequality was proved by Harper. It asserts that the Hamming ball minimizes $A+E$. In that chapter they show how to get a very good asymptotic bound for the same problem.
Well, I actually was aware of the "classical" version with the unit ball instead of $E$ (which is why I am speaking about "an isoperimetric problem"). However, not including zero in $E$ can, potentially, change the things, and I wonder whether anything is known for this specific version of the problem - let alone, about the situation where $A$ consists of even vectors only. Thanks for the exact reference, anyway! – Seva Sep 22 '11 at 10:00