<s>[INST] Surface fitting with convexity requirement Hi all,
Consider a cloud of points in 3D space (x,y,z). The data is well-behaved, once plotted the surface looks like some sort of spheroid. I assumed a form for the fitting function f(x,y,z) = c1 x^2 + c2 y^2 + c3 z^2 + c4 x^2y^2 + ..etc
The coefficients were obtained using a least squares approximation. My only problem is that the surface has some concave portions. Does anyone know how to express a constraint that would generate the coefficients (c1,c2, ...) but ensure convexity at all points?
Thanks a lot!
J
 [/INST] For your apparent purpose, in dimension $n$ it is convenient to begin with a homogeneous polynomial of total degree $2n$ with all individual exponents even. Then, for translates and rotations, all sorts of lower degree and odd exponent terms may show up. 
In $\mathbb R^2,$ a rounded version of an ordinary square is
$$  A(x^4 + y^4) + B x^2 y^2 = 1.$$ The ordinary unit circle is $A=1, B = 2.$ Disjoint hyperbolas are $A=1, B=-2.$ A somewhat squared shape, indeed the $L^4$ "unit circle," is $A=1, B=0.$ An alternative to Piet Hein's "superellipse" is
$A=1, B=1.$ Finally, a real analytic curve that passes through all 8 lattice points with
$ |x| \leq 1,\; |y| \leq 1$ other than the origin itself is $A=1, B = -1,$ or
$$ x^4 - x^2 y^2 + y^4 = 1.$$
At some point I wanted a smooth version of an ordinary cube in $\mathbb R^3,$ meaning that it passed through all
 26 integer lattice points with
$ |x| \leq 1,\; |y| \leq 1,\; |z| \leq 1$ other than the origin itself.
  I wrote
$$  A( x^6 + y^6 + z^6) + B (y^4 z^2 + z^4 x^2 + x^4 y^2 + y^2 z^4 + z^2 x^4 + x^2 y^4) + C x^2 y^2 z^2 = 1.$$
To find $A,B,C$ it is only necessary to check the $(x,y,z)$ triples $(0,0,1),(0,1,1),(1,1,1),$ and evidently
$A=1, B=-\frac{1}{2}, C=1$ works. so
$$  ( x^6 + y^6 + z^6) - \frac{1}{2}(y^4 z^2 + z^4 x^2 + x^4 y^2 + y^2 z^4 + z^2 x^4 + x^2 y^4) +  x^2 y^2 z^2 = 1$$ is a rounded cube. I recall graphing this in spherical coordinates with $\rho$ a function of $\theta, \phi.$ The trouble was that it is very flat near the axes, so without spherical coordinates many different patches were necessary. It is obvious that this is a star-shaped body around the origin, it needs just a little  more work to confirm that it is compact.
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