New estimates for $d_{2,1}$ and $d_{3,2}$

Abstract: Let

K

be a convex body in

m a t h b b R^{n}

. Let

d_{n, n - 1} (K)

be the smallest possible density of a non-separable lattice of translates of

K

. In this paper we prove the estimate

d_{2, 1} (K) l e q f r a c p i s q r t 3 8

for

K s u b s e t m a t h b b R^{2}

, with equality if and only if

K

is an ellipse, which was conjectured by E. Makai. Also we prove the estimate

d_{3, 2} (K) l e q f r a c p i 4 s q r t 3

for

K s u b s e t m a t h b b R^{3}

using projection bodies.

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