Is there a simple way to calculate/estimate the volume of Minkowski sum of an n-dimensional unit ball and an n-dimensional ellipsoid? Even a simple ellipsoid like $\frac{x\_1^2}{a^2} + x\_2^2 + \ldots + x\_n^2 = 1$ will do.



Volume of Minkowski sum of a ball and an ellipsoid
I guess you know that the result can be written as a polynomial.
$$w\_0+w\_1{\cdot}r+\dots+w\_n{\cdot}r^n.$$
So your question is how to estmate the coefficients $w\_i$; this is so called "cross-sectional measures" and they can be defined for any convex body $K$. 


* $w\_0$ is the volume of $K$,
* $w\_n$ is the volume of unit ball.
* $w\_1$ is the area of the boundary of $K$, I think there is no good formula for ellipsoid, but estimates are known and you could write it as an integral.
* I would be surprised if some remaining $w\_i$ can be expressed by simple formula.


If you want to write $w\_i$ as an integral, 
check for example Burago--Zalgaller, Geometric inequalities. 
For example 
$$w\_i=\mathrm{MayBeAConst}\cdot\int\limits\_{\partial K} \sigma\_{i-1}(k\_1,k\_2,\dots,k\_{n-1})\\, d\mathrm{area}.$$
where $\sigma\_i$ is the $i$-th elementary symmetric polynomial and $k\_i$ are principle curvatures.
In your case it is easy to find $k\_i$...


