Let $S$ be a finite set and let $\mathcal{A} \subset\mathcal{P}(S)$ be a family of subsets of $S$. Consider the convex polytope spanned by the characteristic functions of members of $\mathcal{A}$ :
$$C=C\_ \mathcal{A}:=\operatorname{co}\{ \mathbf {1}\_A \, : \, A\in\mathcal{A} \}\, .$$
It's easy to see that every $\mathbf {1} \_ A$, for $A\in\mathcal{A}$, is extremal in $ C\_ \mathcal{A}$ (indeed, if we have a convex combination $\mathbf {1} \_ A= \sum\_{B\in\mathcal{A}}\lambda \_ B\, \mathbf {1} \_ B $, 
then $B\subset A$ for any $B$ corresponding to a coefficient $\lambda \_ B > 0$, so $\sum\_{B\in\mathcal{A}}\lambda \_ B\left(|A|-|B| \right )=0 $, whence $\lambda \_ A =1 $ is the only non-zero coefficient of the convex combination).  

Therefore, the vertex set of $ C\_ \mathcal{A}$ is exactly $\{ \mathbf {1}\_A \, : \, A\in\mathcal{A} \}$, that we may identify abstractly with $\mathcal{A}$ itself.



> 
> **Question 1.** How to describe the complete abstract facial structure of
>  $C$ in terms of the combinatorics of
>  $\mathcal{A}$?
> 
> 
> 


I suspect that for a general family $\mathcal{A}$ this task may prove to be quite hard. If so, I'd like to see known examples of polytopes obtained this way, especially when $\mathcal{A}$ enjoies special regularity properties, such that the skeleton of $C\_ \mathcal{A}$ admits simple description. For instance: 



> 
> Let $S$ be the $r$-th Cartesian power of the set $[n]:=\{1,2,\dots,n \} $ , 
>  $S={n}^r$, and let
>  $$\mathcal{A}:=\{B^{\, r} \, : \, B\subset R \}\, .$$
>  **Question 2.** Which polytope is the corresponding $C\_ n^r:=C \_ \mathcal{A}$? 
> 
> 
> 


The present problem, especially in the latter example, has been suggested to me by a recent interesting [question](https://mathoverflow.net/questions/92511/how-can-one-prescribe-the-pairwise-intersection-measuress-of-n-sets/92590#92590), which is related to the case $r=2$ (the analogous problem of the one described there, where one consider all intersections of $r$ sets extracted from a given family of $n$, yields to the above polytope $C \_ n ^ r\subset \mathbb{R}^{n^r}$). 


**Up-date, April 13, 2012.** Thanks to the very interesting references given so far, I see that my naive suspicions about the difficulty of question **1** were after all right . So, I would like to focus the attention on question **2**: what can be said about $C\_n^r$, at least for $r=2$? Can we at least count the number $f\_k $ of $k$-dimensional faces of $C\_n^2$: which polynomial sequence do they define, $P\_n(x):=\sum\_{k\ge 0} f\_k x^k$ ?



The facial structure of the convex hull of a family of characteristic functions
In addition to the notes by Ziegler referenced in the comment above, there are a few general classes of 0/1 polytopes where one can say something about the facial structure. 


One example is that of the independent set polytope, $P\_{I(M)}$ for a matroid $M$, where one uses the independent sets of the matroid to define the characteristic vectors. In this case, there is a well-known description of the hyperplane description of $P\_{I(M)}$ using the rank function for the matroid. Also, one can describe the facets of the polytope using the combinatorial structure of the matroid. See for example Jon Lee's book titled *A First Course in Combinatorial Optimization*, section 1.7.


Another example of polytopes of this type are permutation polytopes, where one considers the convex hull of a collection of permutation matrices arising as the representation of some finite group. The paper *On Permutation Polytopes*, by Baumeister, Haase, Nill, and Paffenholz, <http://arxiv.org/abs/0709.1615>, contains a nice introduction to this topic and investigations of the type you suggest.


