Geometric properties of some special cones in a Banach space

Let $E$ be a Banach space partially ordered by a cone $K$. Let $B$ be a closed linear operator in $E$ with domain $\Gamma(b)$. In this paper certain cones in the Banach space $\Gamma(b)$ with norm $\|x\|_\Gamma=\|x\|+\|Bx\|$ are singled out for study; a number of their geometric properties are established under the assumption that the cone $K$ in the space $E$ has analogous properties.