Principal kinematic inequalities

Abstract

One hundred years ago, Blaschke and Poincaré derived exact closed-form formulas for integrals over SE(n) of Euler characteristics of intersections of n-dimensional bodies for n = 2, 3. Three decades thereafter Chern extended these formulas to the n-dimensional case. These results form a core tool in the fields of convex geometry, integral geometry, geometric probability, and stochastic geometry. These results, often referred to as “Principal Kinematic Formulae,” are extended here to the case of integrals of set-indicator functions, resulting in inequalities for the case of non-convex bodies. These results are relevant to assessing the frequency of occurrence of collisions that occur in sample-based robot motion planning.