We study endomorphisms constructed by Sarah Koch in her thesis and we focus on the eigenvalues of differential such maps at its fixed points. In Koch's thesis, to each post-critically finite unicritical polynomial, associated a algebraic endomorphism $\mathbb{CP}^k$. showed that differentials along periodic cycles outside post-critical sets have modulus strictly greater than $1$. this article, ...

It is shown that for any sufficiently regular even Minkowski valuation Φ which homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are only solutions to fixed-point problem Φ2K=αK. This significantly generalizes results by Ivaki projection bodies suggests, via Lutwak–Schneider class reduction technique, new approach Petty's conjectured inequality.