We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantity
$$\int_{{{x}_{0}}\in K}{\cdot \cdot \cdot \int_{{{x}_{n}}\in K}{{{\left[ V\left( \text{conv}\left\{ {{x}_{0}},\cdot \cdot \cdot ,{{x}_{n}} \right\} \right) \right]}^{p}}d{{x}_{0}}\cdot \cdot \cdot }\,d{{x}_{n}},n\ge d,p\ge 1}$$
is maximized by triangles among all planar convex bodies
$K$
(parallelograms in the symmetric case). We show that these are the only maximizers, a fact proven by Giannopoulos for
$p\,=\,1$
. Moreover, if
$h:\,{{R}_{+}}\,\to \,{{R}_{+}}$
is a strictly increasing function and
[{{W}_{j}}$
is the
$j$
-th quermassintegral in
${{R}^{d}}$
, we prove that the functional
$$\int_{{{x}_{0}}\in {{K}_{0}}}{\cdot \cdot \cdot }\int_{{{x}_{n}}\in {{K}_{n}}}{h\left( {{W}_{j}}\left( \text{conv}\left\{ {{x}_{0}},\cdot \cdot \cdot ,{{x}_{n}} \right\} \right) \right)}\,d{{x}_{0}}\cdot \cdot \cdot d{{x}_{n}},n\ge d$$
is minimized among the
$(n\,+\,1)$
-tuples of convex bodies of fixed volumes if and only if
${{K}_{0,\,\ldots \,,}}{{K}_{n}}$
are homothetic ellipsoids when
$j\,=\,0$
(extending a result of Groemer) and Euclidean balls with the same center when
$j\,>\,0$
(extending a result of Hartzoulaki and Paouris).