## Stacked Hypercube Vertex Convex Hull Boundary Minimization

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quintopia
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### Stacked Hypercube Vertex Convex Hull Boundary Minimization

This is a simple-to-state problem my friend spent too long explaining to me today about which I know nothing. I thought it might be a good candidate for MMM (Massively Multiplayer Mathematics) of the smart people here. So please, throw out ideas, however small.

Let's say you have a sequence of vertices [imath]v_k, k \in [h], n \le h \le 2^n[/imath] from the n-dimensional {0,1} hypercube. From this we create an [imath]n+1[/imath]-dimensional set [imath]S = conv(<v_k,k>)[/imath], the convex hull of the sequence of vectors with their index appended. Now, we choose a constant r, such that each level set in S, [imath]L_k=\{x \in S:x_{n+1} = k\}[/imath] is contained in an n-dimensional ball, centered appropriately, of radius r.

The problem: For any h, choose the vector sequence [imath]v_k[/imath] so that r is minimized.

Questions: Is there an LP formulation of this problem? (I would guess that such a formulation would be unwieldy or unhelpful, or else this problem would not still be a problem.)
Is there a minimum h below which all solutions are trivial? (It seems to me that if h<2^i, then there is an easy upper bound on r depending on i.)
Are there any greedy-ish heuristic worth trying? Like using consecutive vertices in any hamilton cycle (i.e. a gray code ordering)? (I doubt things are this simple, but stupid ideas are worth taking a stab at first.)

itaibn
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

Can you describe this better? Here is my best translation:

For some n, take an h such the n<=h<=2^n. Pick a sequense of points v_k in a n-dimensional {0,1} hypercube. From this we create an n+1-dimensional set S as the convex closure of all of the <v_k,k>. Now, we choose the minimal constant r, such that each L_k, defined as the set of points x in S such that *something*=k, is contained in an n-dimensional ball of radius r.

The problem: For any n and h, choose the sequence such that r is minimized.

Is there any misunderstanding? and what exactly is xn+1 supposed to mean?
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quintopia
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

You repeated it the same. [imath]x_{n+1}[/imath] is simply the last component of the vector x. What would be better notation for that? Anyway, the point is that it is a level set: The set of points whose height in dimension n+1 is k.

parallax
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

Choose h vertices from a n-dimensional hypercube. Assign each point a "Temperature" T from 1 to h, without replacement. The temperature varies linearly between the points. Minimize the size of all the isothermal surfaces.
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Yakk
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

quintopia wrote:This is a simple-to-state problem my friend spent too long explaining to me today about which I know nothing. I thought it might be a good candidate for MMM (Massively Multiplayer Mathematics) of the smart people here. So please, throw out ideas, however small.

Let's say you have a sequence of vertices [imath]v_k, k \in [h],[/imath]

What is [h]? The set of integers from 1 to h inclusive? 0 to h-1 inclusive? The operator that takes polynomials in h and extracts the linear term?
[imath]n \le h \le 2^n[/imath] from the n-dimensional {0,1} hypercube. From this we create an [imath]n+1[/imath]-dimensional set [imath]S = conv(<v_k,k>)[/imath], the convex hull of the sequence of vectors with their index appended.

<,> usually refers to inner products, so your use is very confusing. I think you are using it to mean <a,b>: (F^n x F) -> F^(n+1) := (a_0, ..., a_n, b)?
Now, we choose a constant r, such that each level set in S, [imath]L_k=\{x \in S:x_{n+1} = k\}[/imath] is contained in an n-dimensional ball, centered appropriately, of radius r.

So L_k is your level set in S. So r measures how 'fat' the level set is at a point is being measured by the r, I suppose, in a round about sense.

The problem: For any h, choose the vector sequence [imath]v_k[/imath] so that r is minimized.

So we want the 'fattest' part of our convex hull to be thin, in the above round sense.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

quintopia
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

Yakk wrote:What is [h]? The set of integers from 1 to h inclusive?

Yes. It's a fairly common shorthand.

Yakk wrote:I think you are using it to mean <a,b>: (F^n x F) -> F^(n+1) := (a_0, ..., a_n, b)?

yes, hence "the sequence of vectors with their index appended." I just made up the notation on the spot, because everything else I thought of would have been more confusing.

parallax
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### Re: Stacked Hypercube Vertex Convex Hull Boundary Minimization

For example, if n=2 and h=4 you assign:
(0,0) -> 1
(1,0) -> 2
(0,1) -> 3
(1,1) -> 4

The level set for 2 goes from (0,0.5) to (1,0). so r is [imath]\sqrt 5 / 2[/imath].

This is the minimal arrangement.
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