PhyslibAlpha.Mathematics.Convex.Choquet.Mixture

For a convex set $S$ in a real vector space $V$, and for any two points $x, y \in S$ and a parameter $t \in [0, 1]$, the mixture is the point $t x + (1 - t) y$. Since $S$ is convex, this point is also an element of $S$.

Let $S$ be a convex set in a real vector space $V$. For any points $x, y \in S$ and any parameter $t$ in the unit interval $[0, 1]$, the mixture $\text{mix}(x, y, t)$ is equal to the vector $t x + (1 - t) y$.

The constant representing the value $\frac{1}{2}$ as an element of the unit interval $[0, 1]$.

The constant $\text{half}$, defined as an element of the unit interval $[0, 1]$, is equal to $1/2$ when coerced to a real number in $\mathbb{R}$.

For a convex set $S$ in a real vector space and a pair of points $p = (x, y) \in S \times S$, this function returns their midpoint, defined as $\frac{1}{2}x + \frac{1}{2}y$. Since $S$ is convex, the resulting midpoint is also an element of $S$.

Let $V$ be a topological vector space over $\mathbb{R}$ (specifically, a topological additive group with continuous scalar multiplication). Let $S \subseteq V$ be a convex set. The function $\text{mixHalf}: S \times S \to S$, which maps a pair of points $(x, y)$ to their midpoint $\frac{1}{2}x + \frac{1}{2}y$, is continuous.

For a function $f: S \to \mathbb{R}$ defined on a convex set $S$ of a real vector space, the property of being a convex function means that for any two points $x, y \in S$ and any parameter $t \in [0, 1]$, the value of the function at the mixture $t x + (1 - t) y$ is at most the mixture of the function's values at $x$ and $y$. That is, the inequality $f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y)$ holds, indicating that the function lies below the chord connecting $(x, f(x))$ and $(y, f(y))$.

Let $V$ be a real topological vector space and $S \subseteq V$ be a convex set. For any continuous linear functional $\ell: V \to \mathbb{R}$, the function $f: S \to \mathbb{R}$ defined by $f(y) = \ell(y)$ is a convex function on $S$. That is, for any points $x, y \in S$ and any $t \in [0, 1]$, the inequality $\ell(tx + (1 - t)y) \le t\ell(x) + (1 - t)\ell(y)$ holds.

For any two distinct points $p$ and $q$ in a convex set $S$, their midpoint $\frac{1}{2}p + \frac{1}{2}q$ is not an extreme point of $S$.

For any vectors $x_1, x_2$ and real scalars $a, b, r$, the difference between the linear combinations $(a + r)x_1 + (b - r)x_2$ and $(a - r)x_1 + (b + r)x_2$ is equal to $2r(x_1 - x_2)$. This identity characterizes the distance between two points positioned at a weight distance $r$ on either side of the point $ax_1 + bx_2$.

Let $S$ be a convex set in a real vector space $V$. If a point $x \in S$ is not an extreme point of $S$, then there exist two distinct points $p, q \in S$ such that $x$ is their midpoint, i.e., $x = \frac{1}{2}p + \frac{1}{2}q$.