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Convex set
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a set of points is convex if it contains every <a href="/facts/Line_segment/V8E0M4qk">line segment</a> between two points in the set. Equivalently, a convex set or a convex region is a set that intersects every <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a> in a <a href="/facts/Line_segment/V8E0M4qk">line segment</a>, single point, or the <a href="/facts/Empty_set/ffEk3eA6">empty set</a>. For example, a solid <a href="/facts/Cube_(geometry)/65lUaAqN">cube</a> is a convex set, but anything that is hollow or has an indent, for example, a <a href="/facts/Crescent/H9oQTpVm">crescent</a> shape, is not convex. </p><p>The <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of a convex set in the plane is always a <a href="/facts/Convex_curve/26Vvp69H">convex curve</a>. The intersection of all the convex sets that contain a given subset A of Euclidean space is called the <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of A. It is the smallest convex set containing A. </p><p>A <a href="/facts/Convex_function/IbzG5SLF">convex function</a> is a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> defined on an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> with the property that its <a href="/facts/Epigraph_(mathematics)/qGYcHa8t">epigraph</a> (the set of points on or above the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of the function) is a convex set. <a href="/facts/Convex_minimization/7D6U4MTN">Convex minimization</a> is a subfield of <a href="/facts/Mathematical_optimization/oRn8Iv5I">optimization</a> that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called <a href="/facts/Convex_analysis/SGhUNLMK">convex analysis</a>. </p><p>Spaces in which convex sets are defined include the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean spaces</a>, the <a href="/facts/Affine_space/tlvI3oX1">affine spaces</a> over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, and certain <a href="/facts/Non-Euclidean_geometry/efKa7De2">non-Euclidean geometries</a>. </p>