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  "body": "\n## Summary\n\nA face of a [[convex set|02AS45T3H9]] $C$ is a convex subset $F$ of $C$ with the following property.\nAny segment of $C$ whose inside meets $F$ lies entirely in $F$.\nIn other words, a face is a convex [[extreme set|WD3RJZC5A7]] of $C$.\nFor a polytope, the nonempty faces are the vertices, edges and higher-dimensional faces, together with the polytope itself.\nThe zero-dimensional faces of $C$ are exactly its [[extreme points|3MSGFJM2NX]].\n\n## Explanations\n\n### L1 human\n\nThe faces of a cube are the cube itself, its six square sides, its twelve edges, its eight corners, and the empty set.\nEach is a flat part of the boundary, or the whole cube, that cannot be entered from outside along a straight line through its inside.\nIf a segment of the cube passes through the middle of an edge, the segment must run along that edge.\nFor a disc, the faces are the empty set, the disc and its single boundary points.\n",
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