Newton's problem of the body of minimum mean resistance

Consider a body $\Omega$ at rest in $d$-dimensional Euclidean space and a homogeneous flow of particles falling on it with unit velocity $v$. The particles do not interact and they collide with the body perfectly elastically. Let $\mathscr R_\Omega(v)$ be the resistance of the body to the flow. The problem of the body of minimum resistance, which goes back to Newton, consists in the minimization of the quantity $(\mathscr R_\Omega(v)\mid v)$ over a prescribed class of bodies.
Assume that one does not know in advance the direction $v$ of the flow or that one measures the resistance repeatedly for various directions of $v$. Of interest in these cases is the problem of the minimization of the mean value of the resistance $\widetilde{\mathscr R}(\Omega) =\displaystyle\int_{S^{d-1}}(\mathscr R_\Omega(v)\mid v)\,dv$. This problem is considered $(\widetilde{\mathrm{P}}_d)$ in the class of bodies of volume 1 and $(\widetilde{\mathrm{P}}{}_d^c)$ in the class of convex bodies of volume 1. The solution of the convex problem $\widetilde{\mathrm{P}}{}_d^c$ is the $d$-dimensional ball. For the non-convex 2-dimensional problem $\widetilde{\mathrm{P}}_2$ the minimum value $\widetilde{\mathscr R}(\Omega)$ is found with accuracy $0.61\%$. The proof of this estimate is carried out with the use of a result related to the Monge problem of mass transfer, which is also solved in this paper. This problem is as follows: find $\displaystyle\inf_{T\in\mathscr T}\int_\Pi\mathrm{f}(\varphi,\tau;T(\varphi,\tau))\,d\mu(\varphi,\tau)$, where $\Pi=[-{\pi}/{2},{\pi}/{2}]\times [0,1]$, $d\mu(\varphi,\tau)=\cos\varphi\,d\varphi\,d\tau$, $\mathrm{f}(\varphi,\tau;\varphi',\tau') =1+\cos(\varphi+\varphi')$, and $\mathscr T$ is the set of one-to-one maps of $\Pi$ onto itself preserving the measure $\mu$.
Another problem under study is the minimization of $\overline{\mathscr R}(\Omega) =\displaystyle\int_{S^{d-1}}|\mathscr R_\Omega(v)|\,dv$. The solution of the convex problem $\overline{\mathrm P}{}_d^c$ and the estimate for the non-convex 2-dimensional problem $\overline{\mathrm P}_2$ obtained in this paper are the same as for the problems $\widetilde{\mathrm P}{}_d^c$ and $\widetilde{\mathrm P}_2$.