If you want to be more formal about this, the buzzword you want to look for in a differential geometry book is the hodge dual (represented by the * operator) which performs exactly the operation you are describing.

The scoop is this: I’m framing an evolutionary search through the set of linear programming algorithms, and for generality’s sake am thinking of starting from a toolkit of simple (flat Euclidean) geometric operations, coupled with a representation of hyperplanes and half-spaces (linear inequality constraints) as n-point tuples.

So now, using the wedge product, we can describe a set of m inequality constraints on n variables as a list of m n-tuples of n-dimensional points. (Yes, there are very many such tuples which can define a given half-space. That’s one of the fun parts about this representation.) And the objective function as an n-dimensional vector.

The rest of the preparation is setting up some simple geometric functions for our evolving solvers to invoke: shortest distances, norms, intersections, vector addition and subtraction, cross products. An identity function, which will compare two tuples with different numerical values represent the same half-space and see if they’re the same onject. A few “maintenance” operators that will rearrange the numerical representation of an object in what might be more “tidy” form (expanding points away from zero and shrinking them down from infinity; centering them; that sort of thing).

The evolving algorithms will be able to perform operations in several dimensions, using a transparent mechanism. They have access to a stack containing, say, points of different cardinality. When they try to AddPoint, they’ll pop the top point off the stack, determine its dimension, and then pull the next point of that same dimension; add the two, and push the result back onto the stack. Lacking a second point, the operation fails gracefully, replacing the original point to the top of the stack.

What you’ve done, I think, allows me to work with these half-spaces as easily. So a hypothetical PointInHalfSpace? function might pop the top point, and the topmost half-space of that cardinality, and push a boolean that states whether the point lies within the half-space.

So — thanks again!

The wedge product v1 ^ v2 ^ v3 ^ … ^ vn defines the n-form (roughly, n-plane) “containing” the vectors v1…vn. In three dimensions, for instance, the wedge product v ^ w of two vectors v and w defines the 2-plane containing them both. The normal vector to that plane is the cross product v x w. In 4-space, v ^ w defines a 2-plane again, but there is no “normal vector” to a plane in 4-space; this is why there’s no cross product.

Now, if you have _four_ points in 4-space, you have three vectors u, v, w, and u ^ v ^ w is a 3-plane, which has a normal vector. Five points in 5-space gives four vectors which wedge into a 4-plane, with a normal vector. We can use these normal vectors to tell which side of the plane a given point is on, just as in 2 and 3 dimensions, and thus we partition the space into two halves. This only works, though, as you point out, if the points are not coplanar – if you try to wedge a vector with a subplane it’s already in, the product is zero.

And, as Wolfgang points out, you can’t partition a general manifold in this way; it works in Euclidean n-space, though.

By the way, I assume you were talking about flat Euclidean space, since in general your problem has no solution. E.g. on the torus a (closed) line does not always divide the manifold in two halfs.