A new test of the convexity of the density support

Given a sample of n independant and identically distributed random vectors drawn from a density f we study two ways of testing the convexity of the support of the density. The first one requires the hypothesis that f is uniform on its support. Two statitistics are proposed, one to test non-convexity due to the boundary, the other one to detect non convexity due to the existance of a hole. p-values for each test can be bounded above but this upper bound depends on unknown parameters. An estimator for this upper bound is given, and it is proved to be “almost surely similar” (with a given convergence speed). It is also proved that there exists a consistent decision rule associated to this test. When the density is unknown the test is adapted via a density estimation with k−nearest neighbors. Other similar results are given. They are obviously weaker (convergence speed can not be given) but still give good results. key words: density-support, convexity, nearest-neighbors.