TY - GEN A1 - Awyong, Poh Wah A1 - Henk, Martin A1 - Scott, Paul R. T1 - Note on lattice-point-free convex bodies N2 - We prove inequalities relating the inradius of a convex body with interior containing no point of the integral lattice, with the volume or surface area of the body. These inequalities are tight and generalize previous results. T3 - ZIB-Report - SC-96-44 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2546 ER - TY - GEN A1 - Henk, Martin A1 - Weismantel, Robert T1 - On Hilbert bases of polyhedral cones N2 - For a polyhedral cone $C=$ pos $\{a^1,\dots,a^m\}\subset R^d$, $a^i\in Z^d$, a subset of integral vectors $H(C)\subset C \cap Z^d$ is called a Hilbert basis of $C$ iff (i) each element of $C\cap Z^d$ can be written as a non-negative integer combination of elements of $H(C)$ and (ii) $H(C)$ has minimal cardinality with respect to all subsets of $C \cap Z^d$ for which (i) holds. We show that various problems related to Hilbert bases are hard in terms of computational complexity. However, if the dimension and the number of elements of the Hilbert basis are fixed, a Hilbert basis can always be computed in polynomial time. Furthermore we introduce a (practical) algorithm for computing the Hilbert basis of a polyhedral cone. The finiteness of this method is deduced from a result about the height of a Hilbert basis which, in particular, improves on former estimates. T3 - ZIB-Report - SC-96-12 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2230 ER -