Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Hyperplane separation theorem
open-in-new
<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the hyperplane separation theorem is a theorem about <a href="/facts/Disjoint_sets/nLr8XRVd">disjoint</a> <a href="/facts/Convex_set/vdAuJRJl">convex sets</a> in <i>n</i>-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. There are several rather similar versions. In one version of the theorem, if both these sets are <a href="/facts/Closed_set/upYnRTUj">closed</a> and at least one of them is <a href="/facts/Compact_set/d0cgXJH7">compact</a>, then there is a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> 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 <a href="/facts/Projection_(set_theory)/10mwpNsM">projections</a> of the convex bodies onto the axis are disjoint. </p><p>The hyperplane separation theorem is due to <a href="/facts/Hermann_Minkowski/tlKrpw69">Hermann Minkowski</a>. The <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach separation theorem</a> generalizes the result to <a href="/facts/Topological_vector_spaces/XRex1ChN">topological vector spaces</a>. </p><p>A related result is the <a href="/facts/Supporting_hyperplane_theorem/RyA2XtaK">supporting hyperplane theorem</a>. </p><p>In the context of <a href="/facts/Support-vector_machine/XobxpdBG">support-vector machines</a>, the <i>optimally separating hyperplane</i> or <i>maximum-margin hyperplane</i> is a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> which separates two <a href="/facts/Convex_hull/D4Id3JDS">convex hulls</a> of points and is <a href="/facts/Equidistant/muTk20sY">equidistant</a> from the two. </p>