  We present an equivalent midpoint locally uniformly rotund (MLUR) renorming
$X$ of $C[0,1]$ on which every weakly compact projection $P$ satisfies the
equation $\|I-P\| = 1+\|P\|$ ($I$ is the identity operator on $X$). As a
consequence we obtain an MLUR space $X$ with the properties D2P, that every
non-empty relatively weakly open subset of its unit ball $B_X$ has diameter 2,
and the LD2P+, that for every slice of $B_X$ and every norm 1 element $x$
inside the slice there is another element $y$ inside the slice of distance as
close to 2 from $x$ as desired. An example of an MLUR space with the D2P, the
LD2P+, and with convex combinations of slices of arbitrary small diameter is
also given.
