Extension complexity of low-dimensional polytopes

Sometimes, it is possible to represent a complicated polytope as projection of much simpler polytope. To quantify this phenomenon, the extension complexity P P defined be minimum number facets (possibly higher-dimensional) from which can obtained (linear) projection. This notion motivated by its relevance combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In paper we study extension complexity parameter general polytopes, more specifically considering families low-dimensional polytopes. First, prove that fixed dimension alttext="d"> d encoding="application/x-tex">d , random -dimensional (obtained convex hull points in ball or on sphere) typically order square root vertices. Second, any cyclic alttext="n"> n encoding="application/x-tex">n -vertex polygon (whose vertices lie circle) at most alttext="24 StartRoot n EndRoot"> 24 encoding="application/x-tex">24\sqrt n . bound tight up constant factor alttext="24"> encoding="application/x-tex">24 Finally, show there exists an alttext="n Superscript o left-parenthesis 1 right-parenthesis"> o ( 1 stretchy="false">) encoding="application/x-tex">n^{o(1)} minus − encoding="application/x-tex">n^{1-o(1)} Our theorems are proved range different techniques, hope will further interest.