Suppose that $\pi:(V\_1,F\_1)\to V\_2$ is a linear surjective map, where $V\_1$ and $V\_2$ are vector spaces and $F\_1$ is a Minkowski norm on $V\_1$. Let $B\_1$ be the unitary ball on $V\_1$. Define $B\_2:=\pi(B\_1)$ and let $\Sigma\_2$ be the board of $B\_2$.


If $\Sigma\_2$ is an indicatrix with respect to some Mnikowski norm on $V\_2$?



How to prove that a subset in a vector space is an indicatrix?
Yes, it is. The projection of the convex body enclosed by $\Sigma\_1$ is a convex body in the space $V\_2$. If you require, as in Finsler geometry, the quadratic convexity, then that is also preserved. In normed spaces this is just the construction of the quotient norm in the quotient of $V\_1$ by the kernel of the projection (you can then identify the quotient with $V\_2$). This construction comes up in the geometry of isometric submersions in Finsler geometry. See my [paper](http://www.ams.org/journals/proc/2001-129-08/S0002-9939-01-05910-X/S0002-9939-01-05910-X.pdf) with Duran.


