A Cevian simplex, for 
,
corresponding to an interior point 
of an 
-simplex 
is the 
-simplex 
whose 
th polytope vertex 
is the intersection
of the line 
with the 
-dimensional face
opposite 
(Aliyev 2026).
If 
has normalized barycentric coordinates 
,
where 
,
then 
has barycentric coordinates
 |
(1)
|
Writing 
for the volume of an 
-simplex, the volume
ratio is
 |
(2)
|
The sharp volume bound is
 |
(3)
|
(Reutter and Leuenberger 1964, Klamkin 1971, Aliyev 2026), with equality iff 
is the geometric
centroid of 
.
The 
-simplex 
is one of the 
component 
-simplexes into which 
partitions the Cevian simplex. Its volume
ratio is
 |
(4)
|
Aliyev (2026) proved the sharp bound
 |
(5)
|
where
 |
(6)
|
Equality holds when 
has barycentric coordinates 
in the indicated order.
For 
,
the component bound reduces to
 |
(7)
|
where 
is the golden ratio. For 
, it becomes
 |
(8)
|
