The exact bound for the reverse isodiametric problem in 3-space

Abstract: Let

K

be a convex body in

m a t h b b R^{3}

. We denote the volume of

K

by

V o l (K)

and the diameter of

K

by

D i a m (K) .

In this paper we prove that there exists a linear bijection

T : m a t h b b R^{3} o m a t h b b R^{3}

such that

V o l (T K) g e q f r a c s q r t 2 12 D i a m (T K)^{3}

with equality if

K

is a simplex, which was conjectured by Ende Makai Jr. As a corollary, we prove that

V o l (K) g e q f r a c 112 o m e g a (K)^{3}

, where

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

is a convex body and

o m e g a (K)

is the lattice width of

K

. In addition, there exists a three-dimensional simplex

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

such that

V o l (D e l t a) = f r a c 112 o m e g a (D e l t a)^{3} .

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