A Lower Bound for the Volume of Strictly Convex Bodies with Many Boundary Lattice

Abstract

In a recent paper it was shown that if a strictly convex body C in n-dimensional space contains N noncoplanar lattice points (i.e., points with integer coordinates) on its boundary, then S(C)>fc(n)JV("+1)/" where SiC) denotes the surface area of C and kin) > 0 is a constant depending only on n [1]. If F(C) denotes the volume of C and ViC)^c'in)[SiC)JK"-1) where c'(«) > 0 is a constant depending only on n (which is true if C is an n-dimensional sphere for example), then the above theorem implies that ViC)>k'in)Nin+1)li"-l) where k\n) > 0 is a constant depending only on n. The object of this paper is to show that the restriction to r,(Q|c'(n)[S(q]"/t""1) is unnecessary. The main result will be the following theorem. Theorem. // C is an n-dimensional strictly convex body with N noncoplanar lattice points on its boundary, then ViQ>Kin)Nin+imn-i) where ;c(n) > 0 depends only on n. This paper will be divided into two sections. In the first section we shall obtain certain elementary but necessary results. In the second section we shall prove the above theorem. (!) The research on this paper was begun at the University of Cambridge under a Fulbright grant and completed at the University of Pennsylvania under a National Science Foundation Graduate Fellowship.

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