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This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated as spectrahedra and their shadows (projections). Several general results obtained in the last 15 years by a variety of different methods are presented in the book, along with the necessary background from algebra and geometry.

Convex domains. --- Matrix inequalities. --- Inequalities (Mathematics) --- Matrices --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Algebraic geometry. --- Convex geometry. --- Discrete geometry. --- Mathematical optimization. --- Algebraic Geometry. --- Convex and Discrete Geometry. --- Optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Discrete mathematics --- Geometry --- Combinatorial geometry --- Algebraic geometry --- Desigualtats matricials --- Dominis convexos