The following are thoughts I had during a joint work with P. Olver on maps from spheres to the convex hull of finitely many points in some finite-dimensional vector space. I do not wish to discuss the main result in that article, but rather an interesting remark related to the definition of these maps. It is a link between convexity and fuzzy logic. I know that the link (or perhaps something stronger like an equivalence of sorts) between convex geometry and probability is quite well known. But in this case, it is a link between convex geometry and fuzzy logic.
In fuzzy logic, instead of just having a statement being either true or false, one allows a statement to be say 72% true and thus 28% false. This applies in particular to the statement "point p belongs to a subset S of some set X". Characteristic functions become functions with values in $[0, 1]$. Moreover, one extends the definitions of AND, OR and so on, to this fuzzy logic setting.
Let us say we are given $n$ distinct points in $\mathbb{R}^2$, say $\mathbf{x}_i$, for $i = 1, \ldots, n$. Let $v \in S^1$ denote a unit vector.
If we think of $v$ as based at $\mathbf{x}_1$ and define $H(\mathbf{x}_1, v)$ to be the closed hyperplane defined by $$ H(\mathbf{x}_1, v) = \{ \mathbf{x} \in \mathbb{R}^2; (\mathbf{x}-\mathbf{x}_1, v) \leq 0 \}, $$ where $(-,-)$ denotes the Euclidean inner product in $\mathbb{R}^3$. For $1 \leq i \leq n$, $i \neq 1$, we would like to assign a truth value (in the sense of fuzzy logic) to the statement: $\mathbf{x}_i$ belongs to $H(\mathbf{x}_1, v)$ (S).
One reasonable choice of truth value is to base the truth value on the angle between $v$ and the normalization of $\mathbf{x}_i - \mathbf{x}_1$. Let $$v_{1i} = \frac{\mathbf{x}_i - \mathbf{x}_1}{\lVert \mathbf{x}_i - \mathbf{x}_1 \rVert}$$.
One may suggest as truth value of the statement (S) the following: $$\operatorname{max}\{0, - (v_{1i}, v)\},$$ which is in $[0, 1]$. It is $0$ if $\mathbf{x}$ does not belong to $H(\mathbf{x}_1, v)$, but if it does, then it is between $0$ and $1$ and it is close to $1$ if $v_{1i}$ is close to minus $v$. It is a reasonable truth value of (S), based only on angles. I am sure it is not the only choice though.
If we define the truth value of the AND of two statements to be simply the product of the two truth values (which extends the usual truth table of AND), the truth value of the statement: $\mathbf{x}_i$ belongs to $H(\mathbf{x}_1, v)$ for any $i$, with $1 \leq i \leq n$ and $i \neq 1$, (T)
becomes
$$ c_1(v) := \prod_{i=2}^n \operatorname{max}\{0, - (v_{1i}, v)\}. $$
Similarly, one may replace $\mathbf{x}_1$ with another one of the $\mathbf{x}_j$, and define similarly $c_j(v)$, for $j = 1, \ldots, n$. Each $c_j$ is thus a piecewise smooth function from $S^1$ to $[0, 1]$ and has the truth value interpretation above.
If, for a given $v \in S^1$, the $c_j(v)$ do not all vanish, it would then make sense to multiply the $c_j(v)$ by a factor so that, after having done so, we get $$ \sum_{i = 1}^n c_j(v) = 1. $$ Note that, by abuse of notation, I kept the same notation for the scaled $c_j(v)$, compared to the initial $c_j(v)$
In the article with P. Olver, we then looked at: $$\sum_{i = 1}^n c_j(v) \mathbf{x}_j$$ and studied the properties of the corresponding map from the set of $v$s for which not all the $c_j(v)$ vanish to the above point in the convex hull of the $\mathbf{x}_i$. It gets analytic and somewhat technical, and I did mention that I did not want to discuss the result we obtained.
But I did want to generate a discussion. Have links between convex geometry and fuzzy logic been studied before, such as what I have mentioned in this post? If so, could someone provide some references please?
Also, I am kind of interested as to what some other possible (reasonable) truth values of (S) could be. This post has a somewhat open-ended aspect to it, but I hope it will generate some interesting comments/answers.
