We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments
$${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$
where x1,...,xm are m vectors in
$$\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$$ .
According to the definition of Firey, the Minkowski p-sum of segments is given by
$$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$
We describe related geometric properties of the Lewis maps associated to classical operator norms.