A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points

Depends on

• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length

Used by

• The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary Corollary
• Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
• Convex and strictly convex functions on Euclidean convex sets Definition
• Star-shaped open subsets of Euclidean space Definition
• Supporting and strictly separating hyperplanes in Euclidean space Definition
• Every cycle in a round annulus has one period, that of the central circle Example
• The fundamental group of the unit interval is trivial at every basepoint Example
• FALSE: every continuous complex-valued function on a convex domain has a primitive False statement
• A bounded separately holomorphic function on a polydisc is Lipschitz on every smaller polydisc Lemma
• A convex set and its closure have the same interior and boundary Lemma
• A disc missing p carries a holomorphic logarithm of z-p Lemma
• Metric projection onto a closed convex set satisfies the variational inequality Lemma
• On a convex open set the difference quotient is an average of the derivative along the segment Lemma
• The filled difference quotient of a holomorphic function is jointly continuous Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• Every sublevel set of a convex function is convex Proposition
• Star-shaped plane domains are homologically simply connected Proposition
• Complex star-shaped and convex domains are the published Euclidean notions under the identification ℂ=ℝ² Remark
• A circle traversed k times has winding number k inside and 0 outside Theorem
• A function is convex exactly when its epigraph is convex Theorem
• A nonconstant scalar holomorphic function on a domain in ℂᵐ is an open map Theorem
• Convex domains are holomorphically convex Theorem
• Disjoint nonempty Euclidean convex sets have a separating hyperplane Theorem
• Every point has a unique nearest point in a nonempty closed Euclidean convex set Theorem
• Multivariable Taylor formula with a Lagrange remainder along a line segment Theorem
• On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂ Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)