  The isodiametric inequality states that the Euclidean ball maximizes the
volume among all convex bodies of a given diameter. We are motivated by a
conjecture of Makai Jr.~on the reverse question: Every convex body has a linear
image whose isodiametric quotient is at least as large as that of a regular
simplex. We relate this reverse isodiametric problem to minimal volume
enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large
volume simplices in any decomposition of the identity matrix.
  As a result, we solve the reverse isodiametric problem for $o$-symmetric
convex bodies and obtain a strong asymptotic bound in the general case. Using
the Cauchy-Binet formula for minors of a product of matrices, we obtain
Dvoretzky-Rogers-type volume bounds which are of independent interest.
