RATIO OF CONVEX BODIES
Let K be a convex body in n-dimensional Euclidean space R" (n i> 2), V(K) > 0 its n-dimensional volume, A(K) its (n-1)-dimensional surface area, and L(K) the number of lattice points (points with integer coordinates) in the interior of K. We will be concerned with obtaining lower bounds for L(K) from lower bounds for the volume/area ratio V(K)/A (K), and upper bounds for V(K)/A (K) from upper bounds for L(K). These questions were first studied explicitly for n=2 by Bender [1] and Hammer [8, 9]. Wills [16] later considered the case of general n >12, and since then various additional results have been obtained by Bender [2], Hadwiger [6], Hammer [10], Odlyzko [13], Reich [14], and Wills [1%19].
Depending on whether one wishes to obtain bounds for V(K)/A(K) in terms ofL (K) or conversely, it is convenient to work either with
defined for m and n integers, m>_ 1, n>2, or with l (e, n) = min {L (K): V (K)/A (K) >1 c~, K = R"}, defined for n ~> 2 and ~ any positive real number.
Let co, be the volume of the n-dimensional unit sphere. We will prove THEOREM. lf m and n are integers, m>~ 1, n~>2, and c~ is a positive real number, then n kco./ n\ co. / 2 and Β’o,n"c~ n > l (% n) > co, nnc~ "-1 (~ --Β½).
(
) COROLLARY. If m >>. 1 isfixed, then tim s (m, n) = Β½. (3) n"* oo If m >1 co,n", then s (m, n) > s (m, n + 1). (4) * During the preparation of this paper the second author was supported by a Hertz Foundation Fellowship.