Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We prove that computing the vertex centroid of an H-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for H-polytopes. We also consider the problem of approximating the vertex centroid by findin...

The motion of a two-dimensional buoyant vortex patch, i.e. patch with uniform density different from the surrounding fluid, is analyzed in terms evolution equations for its centroid, deformation boundary and strength sheet which essential to enforce pressure continuity across boundary. centroid are derived by linear momentum analysis that applying Euler's on boundary, while studied centroid-fix...