Estimations Of Covering Numbers And Functionals Of A Class Of Convex Bodies

Aiming at Hadwiger’s conj ecture that the least number c（K）of translates of the interior of a convex body K needed to cover K in Rⁿ is at most 2ⁿ,according to two facts that c（K）equals the least number of directions needed to illuminate the boundary of K and c（K）equals the least number of smaller nomothetic copies of K needed to cover K,we estimate the upper bound of the covering number and the cover functional of K when K is the convex hull of two parallel non-empty compact convex sets T and B.On the one hand,we study the illumination problem of bdK and prove that the illumination number of T ×{1}（B×{0}）is not more than c（T）（c（B））and c（K）is not more than the sums of c（T）and c（B）,this estimation cannot be improved.Then we prove that c（K）＝c（T）+c（B）=2c（T）if T is a translate of B,we also obtain the relation between c（K）and c（B）if one translate of T is contained in relint B.At last we prove that c（K）＝8 if and only if K is a parallelepiped in R³.Thus Hadwiger’s conjecture is completely solved in R³ for this class of convex bodies.On the other hand,we study the covering functional of K in a special case.When T,B（?）R² are two parallelograms and T=1/2B,we prove thatГ₅（K）=2/3.