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

• Agile online mirror descent Algorithm
• Diagonal AdaGrad for online convex losses Algorithm
• Finite-memory-free proximal bundle method with serious and null steps Algorithm
• Gram-matrix AdaGrad with full or singular diagonal metrics Algorithm
• Projected online gradient descent Algorithm
• Projected subgradient method with finite budget Algorithm
• Uncorrected AMSGrad with a coordinatewise historical maximum Algorithm
• Energy uniqueness for the wave Cauchy problem Corollary
• Minkowski convex-body theorem at equality Corollary
• The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary Corollary
• An isolated boundary point obstructs pointwise-zero Green data Counterexample
• The projected OGD bound does not cover infeasible comparators Counterexample
• Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
• Convex and strictly convex functions on Euclidean convex sets Definition
• Extended-real indicator of a set Definition
• Finite-dimensional convex program, Lagrangian dual, and Slater condition Definition
• Mirror maps, Bregman divergence, and local Hessian norms Definition
• Star-shaped open subsets of Euclidean space Definition
• Successive minima of a convex body Definition
• Supporting and strictly separating hyperplanes in Euclidean space Definition
• A fixed denominator regularizer does not prevent Adam's periodic regret example Example
• A projected gradient step on an interval Example
• A quadratic mirror map gives a projected gradient step Example
• A strictly plurisubharmonic exhaustion of the convex unit ball Example
• AMSGrad retains an earlier second-moment maximum Example
• Constant first-moment weight is outside the proved square-root regret schedule Example
• 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
• The hexagonal A₂ cell: Euclidean cell metric versus graph distance 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
• A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma
• Attained successive minima and adapted flag Lemma
• Bounded primitive integral element for Hermite-Minkowski Lemma
• Compactness makes the Bregman projection well-defined Lemma
• Diagonal AdaGrad subproblem has a unique minimizer Lemma
• Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet Lemma
• Every geometric braid is isotopic to a stacking of signed elementary half twists Lemma

…and 45 more results.

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)