2. [10 points] Consider the set D of vectors (a1, a2, (3, 04) ( R* such that all ai are integers and the sum aj + a2 + a3 + 04 is even. (a) What are the nonzero elements in D at a minimal distance to the origin?(b) Find a collection of inequalities defining the set of all points which are closer to the origin than they are to any other element of D. (Hint: By the Pythagorean theorem, a point x is on the hyperplane midway between y and z if x . (y -2) = =(ly|2 -[z]?).) This describes a set Ap which is a 4-polytope you are familiar with. Explain which one it is and find its vertices. (c) Translating the polytope Ap by each element of D, we obtain the cells of a tessellation of IR (with a cell for each d c D that we may describe as the set of points closer to d than to any other element of D). Give a description of the set C of all vertices of all translations of Ap by de D. (Hint: the point (1,0,0,0) belongs to C.)