The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that
sup
d
(
A
⊕
K
,
B
⊕
L
)
≥
sup
∂
(
A
⊕
K
,
B
⊕
L
)
≥
c
⋅
n
1
−
k
+
k
′
2
n
,
\begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*}
if
K
⊂
R
n
\mathrm {K}\subset \mathbb {R}^n
and
L
⊂
R
k
\mathrm {L}\subset \mathbb {R}^k
are convex and symmetric (the supremum is taken over all symmetric convex bodies
A
⊂
R
n
−
k
\mathrm {A}\subset \mathbb {R}^{n-k}
and
B
⊂
R
n
−
k
′
)
\mathrm {B}\subset \mathbb {R}^{n-k’})
. Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.