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On Asteroid Engineering

I pose the question of maximal Newtonian surface gravity on a homogeneous body of a given mass and volume but with variable shape. In other words, given an amount of malleable material of uniform density, how should one shape it in order for a microscopic creature on its surface to experience the largest possible weight? After evaluating the weight on an arbitrary cylinder, at the axis and at the equator and comparing it to that on a spherical ball, I solve the variational problem to obtain the shape which optimizes the surface gravity in some location. The boundary curve of the corresponding solid of revolution is given by $\ (x^2+z^2)^3-(4\,z)^2=0\ $ or $\ r(θ)=2\sqrt{\cosθ}$, and the maximal weight (at $x=z=0$) exceeds that on a solid sphere by a factor of $\frac35\root3\of5$, which is an increment of $2.6\%$. Finally, the values and the achievable maxima are computed for three other families of shapes.