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Pick a point on each of the positive half-axes in $\mathbb{R}^n$. Put a (unit-norm?) vector at each of the n points.

(a) Is there a hypersurface in the orthant $\mathbb{R}^n_+$ going through these n points with the vectors as outer normals? (The answer ought to be yes, since I'm asking whether there exists a closed=exact 1-form on $\mathbb{R}^n_+$ with given values at a few points.)

(b) Is there a hypersurface with the above properties, and also constrained to have the outer normals at all points lie in some half-space? (I know the half-space in question will contain the n vectors specified above, and also the vector (1,1,...,1).) Equivalently, I'm asking that some other vector points inwards along all of the hypersurface.

(c) I'd quite like the above surface together with the coordinate hyperplanes to enclose a compact set. Is this possible? automatic?

(d) Is there a constructive way to build a hypersurface with properties (a)-(c)? Ideally in the vein of writing down some equations for the surface, or for its normal vector field, in terms of the original points and vectors. (I'd like to build one of those for each orthant and then be able to stitch them together.)

(Motivation: I'm trying to show convergence of a stochastic system by constructing (the level sets of) a Lyapunov function for it. I have quite a lot of freedom for what the function looks like on the interior of each orthant, but it's progressively more and more constrained on all the higher codimension subspaces, until at each half-axis I have a very limited range of possibilities. I'd like to know if these possibilities can be patched together in a sensible way.)

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up vote 4 down vote accepted

This is speculation, not a precise answer, but I wonder if perhaps Minkowski's theorem on the existence of a polytope with prescribed face normals and areas might help? This theorem is described in detail in, for example, Alexandrov's book, Convex Polyhedra, Chapter 7, p.311ff.

The connection to your problem is this. If you specify $k$ unit normals ${\bf n}_i$, you can imagine those determining the orientation of $k$ faces of a convex polytope in $\mathbb{R}^d$. Minkowski's theorem says that, if in addition, you specify face areas $F_i$ such that $\sum_{i=1}^{k} F_i {\bf n}_i = 0$, then there exists a polytope realizing those normals and areas. If I understand your situation correctly, you have some of the ${\bf n}_i$ prespecified. You need to add one more ${\bf n}_i$ (one more to reach $d+1$ in total), as well as choose areas $F_i$, in order to zero that sum. The convexity might(?) then yield the halfspace condition you desire.

This is all discrete, of course, but it should not be difficult to pass from a polytope to a smooth hypersurface.

Example. $d=2$ (my $d$ is your $n$). You specify ${\bf n}_1=(1,0)$ and ${\bf n}_2=(-1,1)$, red below, on "the positive half-axes in $\mathbb{R}^2$" (dashed), and then throw in ${\bf n}_3=(0,-1)$ (green) and appropriate areas, which by Minkowski determine a triangle, which you then approximate by a smooth curve (hypersurface), which has all outer normals in a halfspace.
For general $d$, $d+1$ vectors ${\bf n}_i$ will lead to a simplex. Added. There is a chance that the John ellipsoid in this simplex could serve as a starting point on your other question, "Is there an ellipsoid with given outer normals?"

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Thank you, that's exactly the sort of thing I was hoping for! – Elena Yudovina Oct 16 '11 at 19:38

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