Convex hull of the union of scaled polytopes create few new edges

Let $P$ and $Q$ be two polygons in $mathbb{R}^2$. Given $a > 0$, denote by $aP$ its image under the dilation by $a$ centered around the origin (i.e. the polygon obtained by replacing each vertex $(p_0,p_1)$ by the vertex $(ap_0, ap_1)$). Let $C(P,Q)$ denote the convex hull of the union $P cup Q$, and let $N(P,Q)$ denote the number of new edges created (i.e. edges in $C(P, Q)$ which do not appear in $P$ or $Q$). Let $P+Q$ denote the Minkowski sum of the two polygons.

Based on some empirical results, I was led to the below heuristic (if $X$ is a Gaussian centered at $0$, then $|X|$ follows the half-normal distribution).

Heuristic: Let $a,b$ be chosen from the half-normal distribution, and $P, Q$ be two fixed convex polygons in $mathbb{R}^2$ that are “sufficiently generic”. Then:
$$ mathbb{E}(N(aP, bQ))<10$$

Unfortunately it turns out that this does not hold for all choices of polygons $P$ and $Q$, and finding a counterexample is not difficult (see below for details). Above $10$ is an arbitrarily chosen constant, the key point is that it does not depend on the number of vertices in the polygons $P$ and $Q$. Let a Gaussian random polygon $P$ be the convex hull of $n$ random, independent points in $mathbb{R}^2$ sampled according to the standard normal distribution, for some $n$; here $n$ is the “size” of the polygon. So this leads me to my real question:

Question: What are some conditions which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when $P$ and $Q$ are Gaussian random polygon of fixed size.

In particular, an answer to the above question should rigorously prove that the heuristic holds (with high probability) when $P$ and $Q$ are Gaussian random polygons. The next question is a more general version, with Minkowski sums.

Question 2: Let $a_1, cdots, a_m, b_1, cdots, b_n$ be chosen from a half-normal distribution, and $P_1, cdots, P_m, Q_1, cdots, Q_n$ be polygons in $mathbb{R}^2$. Consider the following statement:
$$ mathbb{E}(N(a_1P_1+cdots+a_mP_m, b_1Q_1+cdots+b_nQ_n)) < 10$$
What are some conditions on these polygons which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when the $P_i$ and the $Q_j$ are Gaussian random polygon of fixed sizes.

If the phrasing of the above questions are too vague, I’d be happy with a proof that the above inequalities hold, with high probability, when the polygons are random Gaussian polygons. One special case I’m interested in is when the sizes of the Gaussian polygons is 2 (i.e. they are line segments).

The counterexample for Q1 can be constructed as follows. Suppose $O P_1 P_2 cdots P_n$ is a convex polygon, with the vertices in clockwise order. Let $P_i P_{i+2}$ intersect $O P_{i+1}$ at the point $Q_{i+1}$. The points should be chosen carefully so that the ratio $frac{OQ_{i+1}}{OP_{i+1}} < epsilon$, where $epsilon>0$ is small (this can be done inductively). Let $P$ be the polygon $OP_2 P_4 …$ and $Q$ the polygon $OP_1 P_3 …$. With this choice, it is easy to see that $mathbb{E}(N(aP, bQ))$ grows linearly with $n$.

mg.metric geometry – Which polytopes can be deformed while keeping their edge-lengths?

Let $PsubsetBbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while

  • keeping its combinatorial type, and
  • keeping its edge-lengths.

I know that the $d$-cube is flexible in this sense.
More generally, most (but not all) zonotopes are flexible (see the comments). Also all polygons are flexible. But are there any others?

$quadquad$

I also know that there are polytopes having several realizations with matching edge-lengths (e.g. see the image here), but these realizations cannot be continuously deformed into each other while preserving all edge-lengths.

polygons – Integer Linear (ILP / MILP) Formulation for Collision Avoidance of Convex Polytopes / Polyhedra

I am looking for a possibility to avoid the collision of two convex polytopes using (mixed integer) linear programming. I know how I can detect a collision (Akgunduz, A., Banerjee, P., and Mehrotra, S. (March 14, 2005). “A Linear Programming Solution for Exact Collision Detection .” ASME. J. Comput. Inf. Sci. Eng. March 2005; 5(1): 48–55. https://doi.org/10.1115/1.1846053):

Vertices of polytopes are given: vj, wj

Weights are variables: αj ⩾ 0, βj ⩾ 0

∑vj αj − ∑wj βj = 0 (1)

∑αj = 1 (2)

∑βj = 1 (3)

Equation (1) applies to each of the x, y, and z coordinates of the vertices for P1 and P2. Here the presence of a feasible solution indicates a collision. If no solution is found, no collision exists. What I am now looking for is the opposite. I am looking for a feasible solution for no-collision and a not feasible solution if the polytopes collide. How can I change the equations to get this? It is also important for me, that the collision avoidance is not included in the objective function, since the objective parameters are others. The equations should just create a feasible solution space for non-collsion solutions.

Thanks in advance.

convex polytopes – Distance to the “boundary” of a polyhedral complex

Suppose I have a polyhedral complex ${P_1, ldots, P_k}$ and let $S := cup_{i = 1}^k P_i$. I am interested in a function which measures the distance from a point $x in S$ to the “boundary” of my polyhedral complex, in other words the lower dimensional faces. More precisely, if $x in text{int}(P_i)$ for some $i$ then my function should return the distance from $x$ to the boundary of $P_i$. Otherwise, $x$ lies on a lower dimensional face and the function should return $0$.

Is there a name for this function? Has it appeared in the literature and been studied before? I’m also wondering if it has any special properties beyond simply being a distance function to a non-convex set. Thanks in advance!

polytopes – Convex helmet of a convex curve as an infinite intersection of the convex helmet of triangles

Leave $ f (x) $ be a univariate convex curve (say $ f (x) = x ^ 2 $) and let the domain be limited. The objective is to demonstrate that the convex hull of this curve in its domain can be expressed as an infinite intersection of the convex hull of the triangles. The convex hull sequence is constructed as follows:

(i) Partition the domain evenly in $ k $ partitions, (ii) construct a triangle in each partition using the two tangents at the end points of the partition and the secant that joins the end points, (iii) take the convex hull of the result $ k $ triangles Let the convex helmet form in the iteration $ k $ be denoted by $ A_k $.

Now we want to demonstrate $ cap_k A_k = operatorname {conv} (y = f (x)) $.
Any help is appreciated.

<img src = "https://i.stack.imgur.com/oDkRt.png" alt = "An example is shown for $ k = 2 $ y $ f (x) = x ^ 2 $ in the domain $ (- 2, 2) $">

Ag algebraic geometry: tests of Euler's formula for n-dimensional dimensioned convex polytopes

Twenty tests of Euler's formula V – E + F = 2 are presented in the geometry scrap yard.

1] Which ones can extend beyond three dimensions to any n-dimensional dimensioned convex polytope?

2] Do you have references to other tests for undefined dimensions?

convex polytopes: deciding whether a subspace intersects with an open polyhedral cone

I am trying to answer the following question:

Leave $ V = text {span} (v_1, …, v_k) subseteq mathbb {R} ^ n $, where $ v_1, …, v_k $ It can be assumed orthonormal. Leave $ C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N } $ be an open polyhedral cone where $ u_1, …, u_N in mathbb {R} ^ n $ They are given. Suppose we have $ v_1, …, v_k $ Y $ u_1, …, u_N $ in hand, how do we decide if $ V cap C = varnothing $ or not?

This is what I tried:

My first approach is to use the separation theorem to try to transform the problem into a standard linear viability problem:
$ V cap C = varnothing $ If and only if $ exist y in mathbb {R} ^ n $ such that
$$
y ^ T (r_1v_1 + cdots + r_kv_k) geq 0, forall (r_1, …, r_k) in mathbb {R} ^ k text {y}
$$

$$
y ^ Tx <0, forall x: x ^ Tu_i <0, i = 1, …, N.
$$

Written more compactly, $ V cap C = varnothing $ If and only if $ exist y in mathbb {R} ^ n $ such that
$$
(i) y ^ TA_V z geq 0, forall z = (z_1, …, z_k) ^ T in mathbb {R} ^ k text {, y}
$$

$$
(ii) y ^ Tx <0, forall x in mathbb {R} ^ n: Ux <0, i = 1, …, N
$$

where $ A_V $ is the $ n times k $ matrix with $ v_1, …, v_k $ like its columns and $ U $ is the $ N times n $ matrix with $ v_1, …, u_N $ as its rows (i) can be treated "sufficiently" considering only $ z = e_1, …, e_k, e_1 + … + e_k $ since they cover positively $ mathbb {R} ^ k $. About (ii), to replace the infinitely many $ x $ finely many $ x $, I would need the extreme rays of $ C $I don't know how to find. But, even if I find these extreme rays, the problem will only become "similar" to the linear feasibility problem. "With some $ <$ conditions ", which I still don't know how to treat completely.

My second approach is simply consider $ V ^ perp $, the orthogonal complement of $ V $. We can use the Gram-Schmidt process to generate an orthonormal basis for $ V ^ perp $tell $ w_1, …, w_ {n-k} $. Leave $ W $ be the $ (n-k) times n $ matrix with $ w_1, …, w_ {n-k} $ as their ranks and $ U $ be the $ N times n $ matrix with $ u_1, …, u_N $ like their ranks then $ V cap C = varnothing $ yes and only if the system
$$
Wx = 0
$$

$$
Ux <0
$$

$$
s.t. x in mathbb {R} ^ n
$$

It has no solution. Again, I don't know how to deal with "$ <$"in $ Ux <0 $.

Sorry for my long paragraph that describes my attempts. Here are my questions:

(1) For a polyhedral cone described by $ C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N } $ with $ u_i $ known, $ i = 1, …, N $Are there simple algorithms or analytical formulas to find their extreme rays?
(2) In the restrictions of a "linear program", if there are any restrictions with "$ <$" instead of "$ leq $"How do I deal with them?

Any ideas, reference books or documents are welcome and appreciated. Thank you.

ag geometry algebraic: polynomial inlays of toric varieties of polytopes?

Background: Leave $ P $ be an integral polytope, and $ X_P $ The toric variety associated with the normal range.

$ X_P $ It is always projective, because the collection of characters corresponding to the points $ mathbb {Z} ^ n cap P $ together they give an inlay of $ X_P $ in projective space.

However, the dimension of this embedding is the number of integer points, which is generally exponentially large in a reasonable description of $ P $.

Questions:

  • Suppose that $ P $ is given by $ Ax leq b $ in $ mathbb {Q} ^ n $, with $ A in M_ {n times m} ( mathbb {Q}) $ Y $ b in mathbb {Q} ^ n $and that they promised us that $ P $ is integral is there a projective incorporation of $ X_P $ that only requires $ POLY (| A |, | b |) $ bits to specify?
  • There is a family $ A_n, b_n $ such that the minimum dimension of an inlay grows exponentially in $ | A |, | b | $?
  • Are there any polytope parameters (in the sense of parametrized complexity) that control the size of a minimum (and efficiently computable) insert?

I'm deliberately lazy about whether the coding of $ A $ is in binary or unary; Inlays of polynomial or pseudopolinomial size would be interesting for me.

Motivation: I am curious about whether there are polytope parameters that are evident through simple additions of the corresponding toric variety, and that could help with computational problems on the side of the polytope.

For example, if we know that $ X_P $ It is a smooth complete intersection and we have the equations that cut it, we can calculate its Euler characteristic using the formula on page 146 of "About Chern's classes and Euler's characteristic for complete non-singular intersections" by Vicente Navarro Aznar. This would count the number of vertices of $ P $, which is usually a $ # P $ Difficult problem Of course, most polytopes will not provide a smooth toric variety or a complete intersection, and it is very likely that calculating the scale is difficult, so this observation is of limited use.

Anyway, I'm curious to know if we can measure the complexity of the politoe by the complexity of the toric variety as a projective variety. The basic question is whether or not we can efficiently find small inlays in general, hence this question.

Linear programming – Formulation of mixed integers of polytopes?

Dice $ t $ different unlimited polyhedra $ P_1: A ^ {(1)} x ^ {(1)} leq b ^ {(1)}, dots, P_t: A ^ {(t)} x ^ {(t)} leq b ^ {(t)} $ we are looking for the representation of $ bigcup_ {i = 1} ^ tP_i $ (not his convex helmet) with mixed integer programming.

  1. What is the standard way of doing it by programming mixed integers with the least number of additional integer variables?

  2. When it is possible to do it with only $ O (t ^ alpha) $ additional whole variables where $ alpha in (0,1) $?

I found a review where they say we can do this with $ O ( log t) $ Whole variables if the polyhedra have a common recession cone.

Are the isomorphic polytopes having the same extension complexity?

Is there a simple proof of the fact that isomorphic polytopes have a complexity of similar extent?