convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of n ...

, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of positive semidefinite matrices forms a

convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . ...

in , and a spectrahedron is a shape that can be formed by intersecting this cone with a linear affine subspace. Spectrahedra are the feasible regions of semidefinite programs. The images of spectrahedra under linear or affine transformations are called ''projected spectrahedra'' or ''spectrahedral shadows''. Every spectrahedral shadow is a

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

that is also semialgebraic, but the converse (conjectured to be true until 2017) is false. An example of a spectrahedron is the spectraplex, defined as :

\mathrm_n = \

where

\mathbf^n_+

is the set of positive semidefinite matrices and

\operatorname(X)

is the trace of the matrix

X

. The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...

References

Real algebraic geometry {{geometry-stub

See also

References