Spectrahedron

File:S asdf rot.png

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of {{math|n × n}} positive semidefinite matrices forms a convex cone in {{math|R^{n × n}}}, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.

Spectrahedra are the feasible regions of semidefinite programs.{{citation|first1=Motakuri|last1=Ramana|first2=A. J.|last2=Goldman|author2-link= Alan J. Goldman |title=Some geometric results in semidefinite programming|journal=Journal of Global Optimization|volume=7|issue=1|pages=33–50|year=1995|doi=10.1007/BF01100204|citeseerx=10.1.1.44.1804}}. The images of spectrahedra under linear or affine transformations are called projected spectrahedra or spectrahedral shadows. Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false.{{Cite journal|last=Scheiderer|first=C.|date=2018-01-01|title=Spectrahedral Shadows|journal=SIAM Journal on Applied Algebra and Geometry|volume=2|pages=26–44|doi=10.1137/17m1118981|doi-access=free}}

An example of a spectrahedron is the spectraplex, defined as

: $\backslash mathrm\{Spect\}\_n\; =\; \backslash \{\; X\; \backslash in\; \backslash mathbf\{S\}^n\_+\; \backslash mid\; \backslash operatorname\{Tr\}(X)\; =\; 1\backslash \}$,

where $\backslash mathbf\{S\}^n\_+$ is the set of {{math|n × n}} positive semidefinite matrices and $\backslash operatorname\{Tr\}(X)$ is the trace of the matrix $X$.{{Cite book|title=Approximation Algorithms and Semidefinite Programming|url=https://archive.org/details/approximationalg00grtn_585|url-access=limited|last=Gärtner|first=Bernd|last2=Matousek|first2=Jiri|publisher=Springer Science and Business Media|year=2012|isbn=978-3642220159|pages=[https://archive.org/details/approximationalg00grtn_585/page/n88 76]}} The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.