Let
Δ
m
=
{
(
t
0
,
…
,
t
m
)
∈
R
m
+
1
:
t
i
≥
0
,
∑
i
=
0
m
t
i
=
1
}
\Delta _m=\{(t_0,\dots , t_m)\in \mathbf {R}^{m+1}: t_i\ge 0, \sum _{i=0}^mt_i=1\}
be the standard
m
m
-dimensional simplex and let
∅
≠
S
⊂
⋃
m
=
1
∞
Δ
m
\varnothing \ne S\subset \bigcup _{m=1}^\infty \Delta _m
. Then a function
h
:
C
→
R
h\colon C\to \mathbf {R}
with domain a convex set in a real vector space is
S
S
-almost convex iff for all
(
t
0
,
…
,
t
m
)
∈
S
(t_0,\dots , t_m)\in S
and
x
0
,
…
,
x
m
∈
C
x_0,\dots , x_m\in C
the inequality
\[
h
(
t
0
x
0
+
⋯
+
t
m
x
m
)
≤
1
+
t
0
h
(
x
0
)
+
⋯
+
t
m
h
(
x
m
)
h(t_0x_0+\dots +t_mx_m)\le 1+ t_0h(x_0)+\cdots +t_mh(x_m)
\]
holds. A detailed study of the properties of
S
S
-almost convex functions is made. If
S
S
contains at least one point that is not a vertex, then an extremal
S
S
-almost convex function
E
S
:
Δ
n
→
R
E_S\colon \Delta _n\to \mathbf {R}
is constructed with the properties that it vanishes on the vertices of
Δ
n
\Delta _n
and if
h
:
Δ
n
→
R
h\colon \Delta _n\to \mathbf {R}
is any bounded
S
S
-almost convex function with
h
(
e
k
)
≤
0
h(e_k)\le 0
on the vertices of
Δ
n
\Delta _n
, then
h
(
x
)
≤
E
S
(
x
)
h(x)\le E_S(x)
for all
x
∈
Δ
n
x\in \Delta _n
. In the special case
S
=
{
(
1
/
(
m
+
1
)
,
…
,
1
/
(
m
+
1
)
)
}
S=\{(1/(m+1),\dotsc , 1/(m+1))\}
, the barycenter of
Δ
m
\Delta _m
, very explicit formulas are given for
E
S
E_S
and
κ
S
(
n
)
=
sup
x
∈
Δ
n
E
S
(
x
)
\kappa _S(n)=\sup _{x\in \Delta _n}E_S(x)
. These are of interest, as
E
S
E_S
and
κ
S
(
n
)
\kappa _S(n)
are extremal in various geometric and analytic inequalities and theorems.