The hit-and-run sampler moves within a full-dimensional compact convex set by choosing a uniform direction, then a uniform point on the chord through the current point in that direction. Its invariant law is the uniform distribution on the set. Starting in the interior keeps the chain in the interior almost surely. To define absolutely continuous transitions also at boundary points, redraw directions whose chords have zero length. Without this convention, a corner can have a positive holding atom.
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