How is the convex body presented to a computing device? It is a polytope or something more general?

If $a_k$ is the mean of the first $k$ elements, then $a_{k+1}=\lceil (ka_k+1)/(k+1)\rceil$. This formula is monotone in $a_k$, hence $a_k$ is a monotone function of $a_1=n$. Hence so is $r(n)$, which equals $a_k$ for all sufficiently large $k$.

I meant you take a product with a fixed infinite-dimensional space and let $X_n$ be the $n$-skeleton of the product.

The limit distance $d(x,y)$ may be zero for some $x\ne y$, so it is not a metric in the usual sense. Do you disallow this, or use a generalized notion of a metric?

Papasoglu estimates the Cheeger constant, which does not control the area bounded be the loop, only the length/area ratio. It can happen that the loop bounds a small area and is very short itself, isn't it?

Apply Morse theorem to $Y=f^{-1}(0)$ being a submanifold of $\mathbb R\times X$, and the function $g(s,x)=s$ restricted to $Y$. This function on $Y$ is proper and regular on, hence Morse theory applies.

Yes the sign was wrong, but it does not matter for the argument.

@Ryan: in the hyperbolic manifold example, how about non-homotopic zero-degree maps?

I answered Question A at mathoverflow.net/questions/136678 (where it was first asked).

In the definition of $A_n$, do you really want to subtract a copy of the denominator from the numerator? This just decreases the value by 1, isn't it?

Can you clarify, perhaps by example, what you have in mind? Of course there is a linear map to $R^1$ such that $f(K)$ is either 0 or the entire line.