Linear Algebra – Hit-n-Run Monte Carlo Scanning on a Convex Polytope

So I'm currently trying to implement an MCMC to evenly sample hyperpoints from the polytope that is defined as $ mathbb {K} = {x in mathbb {R} ^ {n} ; ; text {s.t.} ; ; A , x = b } $ in the special case where there is a generic linear transformation $ A in mathbb {R} ^ {m times n} $. $ b equiv0 in mathbb {R} ^ m $ and the boundary conditions $ 0 leq x leq1 $ stop.

Although I was able to successfully run the simulation (I'm using Julia), there are a few things I'm not sure about:

  • Given that polytopes of this type tend to have star-shaped shapes in larger dimensions, two preprocessing steps are required before starting the simulation:

    1. The first concerns the so-called Blocked river setting That is to find quotation from the text of the exercise, Flux $ i $ so that $ max_ {x in K} x_i = min_ {x in K} x_i = z_i $ and remove such variables from the system by adjusting the vector $ b $, Can someone please explain what the hell that means?

    2. The second is to find an optimal inner point $ mathbb {K} $ as the starting point of the chain, which must be intuitively far from the vertices of the polytope. The text tells me the following: e.g. by calculating $ frac {1} {2n} sum_ {i = 1} ^ n (x ^ { min, i} + x ^ { max, i}) $ Where $ x ^ { min, i} in arg min_ {x in K} x_i $ and $ x ^ { max, i} in arg max_ {x in K} x_i $, Here I simply don't understand the notation: I suppose I should calculate a weighted average of the centers of each edge of the polytope, but I cannot see how this is related to the above wording.

  • as with any coherent mcmc, the walk in the state space must be at t. it satisfies the detailed balance sheetIn this case, the text says that the goal should be distributed $$ p (x) propto delta ^ m (Sx-b) prod_i theta (u_i-x_i) theta (x_i-l_i) $$
    Again, I have no idea how to get this and not how to calculate me.

Each point in a regular polytope has its own antipode point or antipode surface

I apologize for using a non-standard language. When I think about this problem, it seems pretty easy, but it is not.

Maybe it can be rewritten as follows:

There is a unique facet that contains the farthest points from a particular point as the area or point of a particular regular polytope.

How do I prove this result under the theory of Poyltope?

I'm not familiar with the theory of regular polytopes, so I think it's better to recommend the reference.

TO EDIT: Actually, I can prove that to examine every regular polytope. It is possible because I have seen that in higher dimensions there are only three types of regular polytopes. I do not need the proof from the definitions (and the results of the theory) by searching all cases.

abstract algebra – Toric variety associated with the cone over a polytope

To let $ P = [0, m] subset M _ { mathbb {R} ^ 2} $ be the line segment and look over the cone $ P $,

What is the toric variety of the cone over? $ P $?

The thing is, I'm not quite sure how to build a cone $ sigma $ from the polytope $ P $, It should be only the cone of $ me_1 $? In that case we would not receive the variety $ mathbb {C} times T ^ 1 $, Where $ T $ designates the torus?

What does the "affine variety of the cone of a polytope" actually mean? I'm sorry if this question looks too trivial, but I could not find a reference that answered that question explicitly.

Measurement theory – How can a polytope have three different volumes?

I am quite new to geometry and came up with the idea that the same convex polytope can have at least three different volumes.

Consider the permuteder formed by the convex hull of the n! Get points by permuting $ (1, 2, …, n) $, Since the polytope lives on a (n-1) dimensional hyperplane (summation is commutative), it has the volume zero if $ mathbb {R} ^ n $ is the environment space. Fine.

However, an article defines the volume by projecting the vertices down and calculating the volume in (n-1) space. Another volume is obtained by finding the square root of the Gram determinant. But why should these lead to different amounts?

The first one is from Postnikov 2009, Theorem 3.1 and the discussion at the top of page 4. He looks at the generalized permuteder, but that does not change my question.

The second is from Baek and Adams, Prop 2.11.

Intuitively, I'm not sure how the same object can hold different amounts of volume.

I mentioned that I understand how the environment changes things, but beyond $ mathbb {R} ^ n $. $ mathbb {R} ^ {n-1} $I'm not sure why we care about others.

co.combinatorics – The maximum number of Hamilton circuits on a convex polytope embedded in $ mathbb {R} ^ N $

Recently, I wondered if there could be a natural topological complexity measure for convex polytopes embedded in it $ mathbb {R} ^ N $, After some deliberation, it occurred to me that the number of different Hamilton cycles on a convex polytope could be a useful proxy measure. Now suppose that the asymptotic formula for the maximum number of Hamilton cycles on one $ n $vertex convex polytope embedded in $ mathbb {R} ^ N $ is given by:

begin {equation}
f_N (n) tag {1}
end {equation}

I'm curious if there is a polynomial $ P (n) $ so that:

begin {equation}
forall N in mathbb {N}, frac {f_ {N + 1} (n)} {f_ {N} (n)} leq P (n) tag {2}
end {equation}

For the sake of completeness $ (N = 2, n = 4) $ We have a square with 8 H cycles and in the case $ (N = 3, n = 4) $ We have a tetrahedron with 24 H cycles.

Note: After several Google searches, I still do not know if this issue has already been resolved.

convex geometry – John's ellipsoid of a polytope

Suppose that $ X $ is $ mathbb R ^ n $ with a polyhedral norm, ie the unit sphere of $ X $ is a $ n $-dimensional polytope. Suppose the John Ellipsoid of $ X $ is an Euclidean ball that touches every face of the ball $ X $,

Is it true that we can find? $ n $ orthonormal vectors that make up the ball $ X $ Invariant under permutation of its order and character change? (That is, the symmetry group of $[-1,1]^ n $ is a subgroup of the symmetry group of the ball of $ X $?)

co.combinatorics – $ q $ -Kostant Partition Function and Flow Polytope?

It is known that the costant partition function is related to volumes and Ehrhart polynomials of river polytopes of graphene (see, for example, https://link.springer.com/article/10.1007/s00031-008-9019-8 or https: // academic oup.com/imrn/article-abstract/2015/3/830/649778).

There is a natural one $ q $Analysis of the costant partition function; for example, Lusztig & # 39; s $ q $Analysis of the weight multiplicity can be defined in this regard $ q $Costant partition function (see, for example, https://arxiv.org/abs/1406.1453). (Note that for Type A Lusztig & # 39; s $ q $Analogs of weight multiplication are also referred to as Kostka-Foulkes polynomials.

Is there a work to understand the $ q $-analog of the costant partition function in the sense of “$ q $Hard-on theory of flow polytopes?

Smallest box with integer lattice point in a polytope?

Given an integer lattice $ mathcal L subseteq mathbb Z ^ n $ represented by base $ mathcal B $ and an integer linear program $ Ax leq b $ from where $ x in mathbb Z ^ n $ is unknown with $ A in mathbb Z ^ {m times n} $ and $ b in mathbb Z ^ m $ There is a narrow upper limit for $$ min_ {x in mathcal L: ax leq b} prod_ {i = 1} ^ n (1+ | x_i |)? $$

What about the simplest case of $ mathcal L = mathbb Z ^ n $?