DistanceToMeasure

The empirical distance-to-measure function (Chazal, Cohen-Steiner & Merigot, "Geometric inference for probability measures", Foundations of Computational Mathematics 11:733-751, 2011): for a point set X with the uniform empirical measure and mass parameter k (equivalently m0 = k/|X|),

f(x) = ( (1/k) * sum over the k nearest neighbours y of x (including x itself) of d(x,y)^q )^(1/q)

q defaults to 2, matching GUDHI's own gudhi.point_cloud.dtm.DistanceToMeasure default. Self-inclusion (x counts as one of its own k neighbours, at distance 0) is likewise GUDHI's own convention -- confirmed both from that class's own docstring ("k: number of neighbors (possibly including the point itself)") and by reproducing its worked doctest byte-for-byte (DistanceToMeasureSpec; see .claude/WORKLOG-dtm-filtrations.md for the full derivation and every oracle value checked against). One consequence: k = 1 always gives f = 0 everywhere (a point's own nearest neighbour, itself, is at distance 0) -- the degenerate case every consumer of this function (DtmRipsSimplexStream, DTM-weighted alpha.PowerDistance) should reduce to its un-weighted construction at.

Takes a SpatialQuery, not a FiniteMetricSpace alone, so callers choose the k-NN strategy: BruteForce is the safe default (apply's own convenience overload) since VP-tree pruning assumes the triangle inequality, which not every FiniteMetricSpace in this codebase actually satisfies (ExplicitMetricSpace enforces nothing -- GUDHI's own docs feed correlation-derived "distance" matrices through exactly this class). JVPTree is only safe over a genuine metric.