PhyslibAlpha.Mathematics.Geometry.Simplex

Let $E$ be a real vector space and $K \subseteq E$ be a set. Suppose $p, q \in E$ are distinct points ($p \neq q$). If $K$ is the convex hull of $\{p, q\}$ and the set of extreme points of $K$ is $\{p, q\}$, then $K$ is a simplex.

Let $E$ be a finite-dimensional real vector space and $K \subseteq E$ be a simplex. Let $\iota$ be a finite set and $p : \iota \to E$ be an injective function such that for every $i \in \iota$, the point $p(i)$ is an extreme point of $K$. Then the cardinality of $\iota$ is at most the dimension of $E$ plus one: $$|\iota| \le \dim_{\mathbb{R}} E + 1$$

Let $s$ be a finite set of indices and $w_i$ be real-valued weights for $i \in s$. If the sum of the weights is zero, $\sum_{i \in s} w_i = 0$, then the sum of the magnitudes of the negative weights is equal to the sum of the positive weights, expressed as $\sum_{i \in s} \max(-w_i, 0) = \sum_{i \in s} \max(w_i, 0)$.

Let $s$ be a finite set of indices. Let $w_i \in \mathbb{R}$ be weights and $p_i$ be points in a real vector space $E$ for $i \in s$. If the linear combination of these points vanishes, i.e., $\sum_{i \in s} w_i p_i = 0$, then the sum of the points scaled by the positive parts of the weights equals the sum of the points scaled by the absolute values of the negative parts of the weights: $$\sum_{i \in s} \max(w_i, 0) p_i = \sum_{i \in s} \max(-w_i, 0) p_i$$

Let $\{p_i\}_{i \in \iota}$ be a family of points in a real vector space $E$. Suppose that for every finite subset $s \subseteq \iota$ and any weight functions $w_1, w_2: \iota \to \mathbb{R}$, if the weights are non-negative ($\forall i \in s, w_{1,i} \ge 0$ and $w_{2,i} \ge 0$), sum to unity ($\sum_{i \in s} w_{1,i} = 1$ and $\sum_{i \in s} w_{2,i} = 1$), and satisfy $\sum_{i \in s} w_{1,i} p_i = \sum_{i \in s} w_{2,i} p_i$, then the weights must be identical ($w_{1,i} = w_{2,i}$ for all $i \in s$). Under these conditions, the family $\{p_i\}_{i \in \iota}$ is affinely independent.

The standard simplex in the space of functions $\iota \to \mathbb{R}$ is the set of vectors $f$ such that every coordinate is non-negative, $f(i) \ge 0$ for all $i \in \iota$, and the sum of the coordinates is equal to $1$, $\sum_{i \in \iota} f(i) = 1$.

The standard simplex in the space of functions $\iota \to \mathbb{R}$, defined as the set of vectors $f$ such that each coordinate is non-negative ($f(i) \ge 0$ for all $i \in \iota$) and the coordinates sum to one ($\sum_{i \in \iota} f(i) = 1$), is a convex set.

Let $\iota$ be an index set. For any vector $f: \iota \to \mathbb{R}$ in the standard simplex (the set of functions such that $f(i) \ge 0$ for all $i \in \iota$ and $\sum_{i \in \iota} f(i) = 1$), every component $f(i)$ satisfies the inequality $f(i) \le 1$.

For any index $i \in \iota$, the standard basis vector $e_i \in \mathbb{R}^\iota$ (the vector whose $i$-th coordinate is $1$ and all other coordinates are $0$) is an element of the standard simplex in $\mathbb{R}^\iota$.

In the space of functions $\mathbb{R}^\iota$, the convex hull of the range of the standard basis vectors $\{e_i\}_{i \in \iota}$ (where $e_i(j) = 1$ if $i = j$ and $0$ otherwise) is equal to the standard simplex. The standard simplex is the set of all vectors $x \in \mathbb{R}^\iota$ such that each coordinate $x_i$ is non-negative and the sum of all coordinates $\sum_{i \in \iota} x_i$ is equal to $1$.

Let $\iota$ be an index set. Let $\Delta^\iota \subset \mathbb{R}^\iota$ be the standard simplex, defined as the set of vectors $f: \iota \to \mathbb{R}$ such that $f(i) \ge 0$ for all $i \in \iota$ and $\sum_{i \in \iota} f(i) = 1$. The set of extreme points of $\Delta^\iota$ is the set of standard basis vectors $\{e_i\}_{i \in \iota}$, where $e_i(j) = 1$ if $i = j$ and $e_i(j) = 0$ otherwise.

For any index set $\iota$, the family of standard basis vectors $\{e_i\}_{i \in \iota}$ in the space $\mathbb{R}^\iota$, defined such that $e_i(j) = 1$ if $i = j$ and $e_i(j) = 0$ otherwise, is affinely independent over the real numbers $\mathbb{R}$.

The standard simplex in $\mathbb{R}^\iota$, defined as the set of vectors $f$ such that $f(i) \ge 0$ for all $i \in \iota$ and $\sum_{i \in \iota} f(i) = 1$, is a simplex. In this context, a simplex is a convex set that is the convex hull of a finite set of affinely independent points.

Let $E$ and $F$ be real vector spaces and $e : E \simeqᵃ[\mathbb{R}] F$ be an affine equivalence. For any subset $s \subseteq E$, let $\operatorname{ext}(s)$ denote the set of extreme points of $s$. Then the image of the extreme points of $s$ under $e$ is equal to the set of extreme points of the image of $s$ under $e$: $$e(\operatorname{ext}(s)) = \operatorname{ext}(e(s))$$ This signifies that an affine equivalence maps extreme points to extreme points.

Let $E$ and $F$ be real vector spaces. If $K \subseteq E$ is a simplex and $e : E \simeqᵃ_{\mathbb{R}} F$ is an affine equivalence, then the image $e(K)$ is also a simplex in $F$.

Let $E$ be a real vector space and $K \subseteq E$ be a subset. Suppose there exists an affine equivalence $e : E \simeqᵃ[\mathbb{R}] (\iota \to \mathbb{R})$ such that the image of $K$ under $e$ is the standard simplex in $\mathbb{R}^\iota$, denoted by $\Delta^\iota = \{ f : \iota \to \mathbb{R} \mid \forall i, f(i) \ge 0 \text{ and } \sum_{i \in \iota} f(i) = 1 \}$. Then $K$ is a simplex.

The unit interval $[0, 1] \subset \mathbb{R}$ is a simplex.