Sum­mary: I am going to ask a ques­tion about what seems to be sim­ple dif­fer­en­tial geom­e­try. It may be just about lin­ear alge­bra. Me, I was a biol­o­gist for too long to remem­ber any of this, alas. But I sup­pose since I’m a stu­dent these days, I should point out that it’s not for home­work — it’s for an amus­ing project in genetic pro­gram­ming. But I really don’t have the math­e­mat­ics to answer it with any cer­tainty. So… any­body? help?

Say we have two points in two dimen­sions. Call them A and B. Together, A and B suf­fice to uniquely define a line, as long as they’re not coin­ci­dent. As an ordered pair, (A,B) can be used to define a half-​​space in the plane: we can just, for instance, take the half of the plane that lies counter-​​clockwise from the vec­tor con­nect­ing A to B. Pos­i­tive axes, and all that. The line pass­ing through AB is a hyper­plane of dimen­sion 1 lying in 2-​​space.

OK. Same drill in three-​​space. Three points A, B and C uniquely define a plane (which is itself 2-​​dimensional), as long as they’re not co-​​linear. We can take the ordered triple (A,B,C) and apply the right-​​hand rule to pick one side of the plane, and define a half-​​space.

You know where we’re going. Four dimen­sions. Four points uniquely define a 3D hyper­plane, as long as they’re not copla­nar (I sup­pose). That’s OK. Gen­er­ally, in n dimen­sions, n points will uniquely define an n–1 dimen­sional hyper­plane, as long as they’re lin­early inde­pen­dent. But. But.

Does the ordered tuple (A,B,C,D) suf­fice to denote a unique half-​​space? I can see how it might, and at the same time I can see how it might not. Because there’s no clear def­i­n­i­tion of the cross prod­uct in 4D, so… what’s the equiv­a­lent of the right-​​hand rule? The top-​​hand rule, which will point in the anawards direc­tion when you have your fore­fin­ger point­ing in the first direc­tion, your mid­dle fin­ger in the sec­ond direc­tion, and your ring fin­ger point­ing in the third direction?

Ow. Oh, cool! My top thumb… like, dis­ap­peared. And here it is back again. Hunh.

Any­way.

Really, all I want to be able to do is come up with an unam­bigu­ous method for defin­ing a spe­cific half-​​space in n dimen­sions using n lin­early inde­pen­dent points, (x₁,x₂,x₃…x_n). And I want the ordered tuple to be the only cue we use. Does it suffice?

If so, is it just the direc­tion orthog­o­nal to the all rays (x₁,x₂), (x₂,x₃), (x₃,x₄) … and (x_n-​​1,x_n)?