Delaunay Meshes 

Create a Delaunay mesh from a set of points.

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X DelaunayMesh[RandomReal[1, {20, 2}]]

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The same can be done in 1D or 3D.

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X DelaunayMesh[RandomReal[1, {10, 1}]]

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X DelaunayMesh[RandomReal[1, {30, 3}]]

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A Delaunay mesh is defined by the property that for every triangle, the circumcircle of the triangle contains none of the other input points.

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X SeedRandom[1234]; \[ScriptCapitalR] = DelaunayMesh[RandomReal[1, {8, 2}]]; circles = Circumsphere @@@ Normal@GraphicsComplex[MeshCoordinates[\[ScriptCapitalR]], MeshCells[\[ScriptCapitalR], 2]]; Show[HighlightMesh[\[ScriptCapitalR], {Style[0, Directive[PointSize[Large], Red]], Style[1, Gray], Style[2, LightGray]}], Graphics[{Orange, circles}], PlotRange -> {{0, 1}, {0, 1}}]

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