Two ovaloids
S
,
S
¯
S,\bar S
can be mapped diffeomorphically onto each other by equal inner normals. If, under this mapping, principal directions are preserved and
\[
[
(
p
−
p
¯
)
−
(
c
k
1
−
1
−
k
¯
1
−
1
)
]
[
(
p
−
p
¯
)
−
(
c
k
2
−
1
−
k
¯
2
−
1
)
]
≤
0
[(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0
\]
everywhere on the unit sphere for a certain constant
c
c
, then
c
=
1
c = 1
and
p
−
p
¯
p - \bar p
= constant. Here
p
,
p
¯
p,\bar p
are the support functions,
k
1
{k_1}
, and
k
2
{k_2}
the principal curvatures of
S
,
k
¯
1
S,{\bar k_1}
and
k
¯
2
{\bar k_2}
the corresponding principal curvatures of
S
¯
\bar S
. Various characterizations of the sphere are obtained as corollaries.