Testing for complete spatial randomness on three dimensional bounded convex shapes

There is currently a gap in theory for point patterns that lie on the surface of objects, with researchers focusing Euclidean space, typically planar and spatial data. Methodology data thus relies geometry therefore inappropriate analysis observed non-Euclidean spaces. Recently, there has been extensions to sphere, however, many other shapes are left unexplored. This part due challenge defining notion stationarity process existing such space lack rotational translational isometries. Here, we construct functional summary statistics Poisson processes defined convex three dimensions. Using Mapping Theorem, can be transformed from any shape unit sphere which symmetries allow constructed. We present first second order properties demonstrate how they used test determine whether an pattern exhibits complete randomness or preference original space. compare this statistic one constructed analogue L-function inhomogeneous sphere. A study Type I II errors our explored through simulations ellipsoids varying