https://hal-mines-paristech.archives-ouvertes.fr/hal-00709210Lantuéjoul, ChristianChristianLantuéjoulGEOSCIENCES - Centre de Géosciences - MINES ParisTech - École nationale supérieure des mines de Paris - PSL - Université Paris sciences et lettresEmery, XavierXavierEmeryGeometric Covariograms, Indicator Variograms and Boundaries of Planar Closed SetsHAL CCSD2011compact setstationary random closed setset boundarysingular pointvertexcusp[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST][STAT.TH] Statistics [stat]/Statistics Theory [stat.TH]Lantuéjoul, Christian2012-06-18 11:06:482021-11-17 12:31:142012-06-18 11:06:48enJournal articles10.1007/s11004-011-9364-31In the plane, consider a compact set with nonempty interior and piecewise C1 boundary such that any line segment crosses this boundary at only finitely many points. Then the second derivatives of the geometric covariogram of the compact set at the origin are related to geometrical characteristics of the set boundary. Specifically, the mean second derivative is zero when the boundary has no singular point, it is positive when the boundary has finitely many vertices but no cusp, and it is infinite when the boundary has at least one cusp. Similar results hold for a stationary random closed set with nonempty interior and piecewise C1 boundary such that any line segment crosses this boundary at almost surely finitely many points. For a set with almost surely no singular boundary point, the mean second derivative of its indicator variogram at the origin is zero. In contrast, when there is a positive probability that, in a given bounded domain of the plane, the boundary has finitely many singular points, this mean second derivative is negative, finite if all the singular points are vertices and infinite if some singular points are cusps.