Konvexnost funkce
φ
{\displaystyle \varphi }
[
a
,
b
]
{\displaystyle \left[a,b\right]}
(
∀
x
,
y
∈
[
a
,
b
]
,
x
<
y
)
(
∀
λ
∈
[
0
,
1
]
)
(
φ
(
λ
y
+
(
1
−
λ
)
x
)
≤
λ
φ
(
y
)
+
(
1
−
λ
)
φ
(
x
)
)
{\displaystyle \left(\forall x,y\in \left[a,b\right],x<y\right)\left(\forall \lambda \in \left[0,1\right]\right)\left(\varphi \left(\lambda y+(1-\lambda )x\right)\leq \lambda \varphi (y)+(1-\lambda )\varphi (x)\right)}
Vlastní důkaz proběhne matematickou indukcí podle
n
{\displaystyle n}
n
=
1
{\displaystyle n=1}
n
=
2
{\displaystyle n=2}
n
→
n
+
1
{\displaystyle n\to n+1}
Indukční předpoklad:
φ
(
∑
i
=
1
n
λ
i
x
i
)
≤
∑
i
=
1
n
λ
i
φ
(
x
i
)
{\displaystyle \varphi \left(\sum _{i=1}^{n}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n}\lambda _{i}\varphi \left(x_{i}\right)}
n
+
1
{\displaystyle n+1}
φ
(
∑
i
=
1
n
+
1
λ
i
x
i
)
≤
∑
i
=
1
n
+
1
λ
i
φ
(
x
i
)
{\displaystyle \varphi \left(\sum _{i=1}^{n+1}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n+1}\lambda _{i}\varphi \left(x_{i}\right)}
(
∃
i
0
∈
n
+
1
^
)
(
λ
i
0
≠
1
)
{\displaystyle \left(\exists i_{0}\in {\widehat {n+1}}\right)\left(\lambda _{i_{0}}\neq 1\right)}
(
∀
i
0
∈
n
+
1
^
)
(
λ
i
0
=
1
)
{\displaystyle \left(\forall i_{0}\in {\widehat {n+1}}\right)\left(\lambda _{i_{0}}=1\right)}
∑
i
=
1
n
+
1
λ
i
=
n
+
1
≥
2
{\displaystyle \sum _{i=1}^{n+1}\lambda _{i}=n+1\geq 2}
Protože platí:
∑
i
=
1
n
+
1
λ
i
=
1
{\displaystyle \sum _{i=1}^{n+1}\lambda _{i}=1}
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
=
λ
{\displaystyle \sum _{i=1,i\neq i_{0}}^{n+1}\lambda _{i}=\lambda }
λ
:=
1
−
λ
i
0
{\displaystyle \lambda :=1-\lambda _{i_{0}}}
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
=
1
{\displaystyle \sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}=1}
(
∀
i
∈
n
+
1
^
,
i
≠
i
0
)
(
λ
i
λ
∈
[
0
,
1
]
)
{\displaystyle \left(\forall i\in {\widehat {n+1}},i\neq i_{0}\right)\left({\frac {\lambda _{i}}{\lambda }}\in \left[0,1\right]\right)}
(
∀
i
∈
n
+
1
^
,
i
≠
i
0
)
(
λ
i
∈
[
0
,
λ
]
)
{\displaystyle \left(\forall i\in {\widehat {n+1}},i\neq i_{0}\right)\left(\lambda _{i}\in \left[0,\lambda \right]\right)}
φ
(
∑
i
=
1
n
+
1
λ
i
x
i
)
=
φ
(
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
x
i
+
λ
i
0
x
i
0
)
=
φ
(
λ
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
x
i
+
(
1
−
λ
)
x
i
0
)
{\displaystyle \varphi \left(\sum _{i=1}^{n+1}\lambda _{i}x_{i}\right)=\varphi \left(\sum _{i=1,i\neq i_{0}}^{n+1}\lambda _{i}x_{i}+\lambda _{i_{0}}x_{i_{0}}\right)=\varphi \left(\lambda \sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}x_{i}+(1-\lambda )x_{i_{0}}\right)}
y
:=
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
x
i
{\displaystyle y:=\sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}x_{i}}
y
∈
[
a
,
b
]
{\displaystyle y\in \left[a,b\right]}
(
∀
i
∈
n
+
1
^
)
(
x
i
∈
[
a
,
b
]
)
{\displaystyle \left(\forall i\in {\widehat {n+1}}\right)\left(x_{i}\in \left[a,b\right]\right)}
y
{\displaystyle y}
x
i
{\displaystyle x_{i}}
i
{\displaystyle i}
b
{\displaystyle b}
a
{\displaystyle a}
b
=
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
b
{\displaystyle b=\sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}b}
a
{\displaystyle a}
φ
(
λ
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
x
i
+
(
1
−
λ
)
x
i
0
)
=
φ
(
λ
y
+
(
1
−
λ
)
x
i
0
)
{\displaystyle \varphi \left(\lambda \sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}x_{i}+(1-\lambda )x_{i_{0}}\right)=\varphi \left(\lambda y+(1-\lambda )x_{i_{0}}\right)}
φ
(
λ
y
+
(
1
−
λ
)
x
i
0
)
≤
λ
φ
(
y
)
+
(
1
−
λ
)
φ
(
x
i
0
)
{\displaystyle \varphi \left(\lambda y+(1-\lambda )x_{i_{0}}\right)\leq \lambda \varphi (y)+(1-\lambda )\varphi (x_{i_{0}})}
λ
φ
(
y
)
+
(
1
−
λ
)
φ
(
x
i
0
)
=
λ
φ
(
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
x
i
)
+
(
1
−
λ
)
φ
(
x
i
0
)
≤
λ
∑
i
=
1
,
i
≠
i
0
n
+
1
λ
i
λ
φ
(
x
i
)
+
(
1
−
λ
)
φ
(
x
i
0
)
=
∑
i
=
1
n
+
1
λ
i
φ
(
x
i
)
{\displaystyle \lambda \varphi (y)+(1-\lambda )\varphi (x_{i_{0}})=\lambda \varphi (\sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}x_{i})+(1-\lambda )\varphi (x_{i_{0}})\leq \lambda \sum _{i=1,i\neq i_{0}}^{n+1}{\frac {\lambda _{i}}{\lambda }}\varphi (x_{i})+(1-\lambda )\varphi (x_{i_{0}})=\sum _{i=1}^{n+1}\lambda _{i}\varphi \left(x_{i}\right)}
φ
(
∑
i
=
1
n
+
1
λ
i
x
i
)
≤
∑
i
=
1
n
+
1
λ
i
φ
(
x
i
)
{\displaystyle \varphi \left(\sum _{i=1}^{n+1}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n+1}\lambda _{i}\varphi \left(x_{i}\right)}
![{\displaystyle \left[a,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30926fb280a9fdf66fd931e14d4363cb824feaa)
![{\displaystyle \left(\forall i\in {\hat {n}}\right)\left(x_{i}\in \left[a,b\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3f54a59939f4ccbb5ca02e7dae8cd37a8c0039)
![{\displaystyle \left(\forall i\in {\hat {n}}\right)\left(\lambda _{i}\in \left[0,1\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0708a96d7f10447d85ce2834840aa28c4356c6d)
![{\displaystyle \left(\forall x,y\in \left[a,b\right],x<y\right)\left(\forall \lambda \in \left[0,1\right]\right)\left(\varphi \left(\lambda y+(1-\lambda )x\right)\leq \lambda \varphi (y)+(1-\lambda )\varphi (x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe61f8cc89d161017dfed730bdc64d18255cbab5)
![{\displaystyle \left(\forall i\in {\widehat {n+1}},i\neq i_{0}\right)\left({\frac {\lambda _{i}}{\lambda }}\in \left[0,1\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338776e542d90fa3c1c835a1c155ef5a7915c747)
![{\displaystyle \left(\forall i\in {\widehat {n+1}},i\neq i_{0}\right)\left(\lambda _{i}\in \left[0,\lambda \right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e3ed65096cc37052ce002b3d4a72464f79df1d)
![{\displaystyle y\in \left[a,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140ce1e5c159ceddccac7325e6e09042a4f697b4)
![{\displaystyle \left(\forall i\in {\widehat {n+1}}\right)\left(x_{i}\in \left[a,b\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b8013a266e3260c8f1ca59b29fd52ae6a9df73)
