La “conjecture des cylindres”

The “cylinder conjecture” is to suppose that, if K is a gauge, the critical constants of C(K) = K ×] ? 1, +1 ? Rⁿ⁺¹ and of its basis K ? Rⁿ are equal. The connection with packing constants is studied. The concept of Za(ssenhaus)-packing is introduced. ⊕_i=1^hG + (i ? 1)a (G a lattice) is a linear h-lattice, $\text{ζ}{}_{\mathrm{h}}\text{′(K),}\text{ζ}\text{′}{}_{\mathrm{h}}\text{(K)}$, $\text{η}{}_{\mathrm{h}}\text{′(K),}\text{η}\text{′}{}_{\mathrm{h}}\text{(K)}$ the maximum density for translates of K by a linear h-lattice if the translates form a Za-packing for ζ, a packing for η, and if this packing is strict for ^. For K a bounded central star body, it is possible to find H with ζ₁(C(K)) ≤ 2 ζ_H′(K). H is precised for K a gauge and for K = Bⁿ. It is proved by Woods' methods that $\text{η}{}_1\text{(C(B}{}^4\text{)) =}\text{sup}{}_{\mathrm{i=1,\; 3,\; 4,\; 6,}}\text{7}\text{η}{}_{\mathrm{i}}\text{(B}{}^4\text{)}$; a result of Cleaver is used.