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We study the iterations of a class of curvature image operators Λpφintroduced by Ivaki in [J. Funct. Anal. 271 (2016) 2133–2165]. The fixed points of these operators are the solutions of the LpMinkowski 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.

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 $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$

Employing the inverse function theorem on Banach spaces, we prove that in a C2(Sn−1)‐neighborhood of the unit ball, the only solutions of Π2K=cKare origin‐centered ellipsoids. Here Kis an n‐dimensional convex body, ΠKis the projection body of Kand Π2K=Π(ΠK).