## 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!

## Is it possible to emulate common polyhedral dice rolls using just a d6, and if so how?

This is a question that I have asked myself several times, but I have never obtained a really satisfactory result.

The problem is this: Suppose we only have one or more d6 dice (possibly the most common type of dice outside of the pen and paper), but we still want to play D&D 5e or another RPG. The game doesn't really matter here, we just have to be able to emulate different types of dice, like d4, d8, d10, d12 or d20. I suppose that if it is possible to calculate these dice from the rolls of a d6, any other potentially necessary dice rolls can also be calculated in a similar way.

So: How can you emulate the probability results of a d4, d6, d8, d10, d12, and d20 rolling only with a d6?

## 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.