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We study the iterations of a class of curvature image operators Λpφ introduced by Ivaki in [J. Funct. Anal. 271 (2016) 2133–2165]. The fixed points of these operators are the solutions of the Lp Minkowski problems with the positive continuous prescribed data φ. One of our results states that if p∈(−n,1) and φ is even, or if p∈(−n,−n+1], then the iterations of these operators applied to suitable convex bodies sequentially converge in the Hausdorff distance to fixed points. [ABSTRACT FROM AUTHOR]

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$-neighborhood of the unit ball, the only solutions of $\unicode[STIX]{x1D6F1}^{2}K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$-dimensional convex body, $\unicode[STIX]{x1D6F1}K$ is the projection body of $K$ and $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$ [ABSTRACT FROM PUBLISHER]