English
The open segment between x and y consists of all strict convex combinations a x + b y with a,b > 0 and a+b=1.
Русский
Открытый сегмент между x и y состоит из всех строгих выпуклых комбинаций a x + b y с a,b > 0 и a+b=1.
LaTeX
$$$openSegment \\; 𝕜\\; x\\; y = \\{ z \\in E : \\exists a,b: 𝕜, 0 < a, 0 < b, a+b=1, a\\cdot x + b\\cdot y = z \\}$$$
Lean4
/-- Open segment in a vector space. Note that `openSegment 𝕜 x x = {x}` instead of being `∅` when
the base semiring has some element between `0` and `1`.
Denoted as `[x -[𝕜] y]` within the `Convex` namespace. -/
def openSegment (x y : E) : Set E :=
{z : E | ∃ a b : 𝕜, 0 < a ∧ 0 < b ∧ a + b = 1 ∧ a • x + b • y = z}
