4:["$","div",null,{"className":"min-h-screen bg-white flex flex-col","children":[["$","div",null,{"className":"px-6 pt-6 pb-4 border-b border-gray-100","children":[["$","div",null,{"className":"flex items-center gap-1.5 text-xs text-gray-400 mb-4","children":[["$","a",null,{"href":"/","className":"hover:text-[#26a69a] transition-colors","children":"Market"}],["$","span",null,{"children":"›"}],["$","span",null,{"className":"text-gray-600","children":"Polyhedral complex"}]]}],["$","div",null,{"className":"text-[10px] text-gray-400 uppercase tracking-widest mb-1","children":"Company Profile"}],["$","h1",null,{"className":"text-2xl font-bold text-gray-900","children":"Polyhedral complex"}],null]}],["$","div",null,{"className":"flex-1 px-6 py-6","children":[["$","p",null,{"className":"text-sm text-gray-600 leading-relaxed mb-8","children":"In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements."}],["$","div",null,{"className":"columns-1 md:columns-2 gap-10","children":[["$","div","0",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Definition"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"A polyhedral complex \\mathcal{K} is a set of polyhedra that satisfies the following conditions: :1. Every face of a polyhedron from \\mathcal{K} is also in \\mathcal{K}. :2. The intersection of any two polyhedra \\sigma_1, \\sigma_2 \\in \\mathcal{K} is a face of both \\sigma_1 and \\sigma_2. Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in \\mathcal{K} may be empty. ==Examples=="}}]]}],["$","div","1",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Examples"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"• Tropical varieties are polyhedral complexes satisfying a certain balancing condition. • Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex. • Voronoi diagrams. • Splines. ==Fans=="}}]]}],["$","div","2",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Fans"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"A (polyhedral) fan is a polyhedral complex in which every polyhedron is a cone from the origin. Examples of fans include: • The normal fan of a polytope. • The fan associated to a toric variety (see ). • The Gröbner fan of an ideal of a polynomial ring. • A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation. • The recession fan of a tropical variety. == References =="}}]]}]]}],"$Lf"]}],"$L10"]}]