Everything About ___
Live briefing

Everything about Hyperplane separation theorem

On the existence of hyperplanes separating disjoint convex sets

In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Hyperplane separation theoremA 5-minute orientation
  2. Skim a recent paper: Separation axioms, Urysohn’s Lemma and Tietze Extention Theorem for extended pseudo-quasi-semi metric spacesInfluential work since 2022
  3. Pick up a classic book: Calculus, Early TranscendentalsThe most-reprinted book on the subject
  4. Read the landmark paper: On the Separation Theorem of Stochastic ControlCited 452 times
  5. Go sideways: Glossary of functional analysisA closely related topic
  6. Go sideways: Numerical rangeA closely related topic

đź“– Definitions

No dictionary entry for this exact phrase.

🗂️ The subject at a glance

  1. Statements and proof
  2. Case with possible intersections
  3. Converse of theorem
  4. Counterexamples and uniqueness
  5. More variants
  6. Use in collision detection

🏆 Landmark papers (most cited)

🆕 Influential since 2022

đź“š Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

đź”— Related topics