Sausages are good packings

AbstractLetBd be thed-dimensional unit ball and, for an integern, letCn={x1,...,xn} be a packing set forBd, i.e.,|xi−xj|≥2, 1≤i<j≤n. We show that for every % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hCaeHbmv3yPrwyGmvyUnhiv5wAJ9gzLbac% gaGaa4hpamaakaaabaaceaGaa0NmaaWcbeaaaaa!4471! $$p< \sqrt 2 $$ a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(Cn)+ρBd)≥V(conv(Sn)+ρBd), whereSn is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d−2,V(conv(Cn)+ρK)≥V(conv(Sn)+ρK), whereCn is a packing set with respect toK andSn is a minimal “sausage” arrangement ofK, holds.