Guarantee strict convexity at a point
$begingroup$
Suppose we have a univariate function $f(t), tin [0,1]$. We define
begin{align}
G(p) = sup_{tin [0,1]} p f(t) + (1-p) f(1-t), text{ for } pin [0,1].
end{align}
Clearly $G(p)$ is a convex function in $p$ since it is the supremum of a family of convex functions in $p$. However, we want a stronger requirement: we would like to make sure that $G(p)$ is strictly convex at point $p = 1/2$. In other words, we want to guarantee that for any $x,yin [0,1], x neq 1/2, yneq 1/2$, any $uin (0,1)$ such that $u x + (1-u) y = 1/2$, we have
begin{align}
G(1/2) < u G(x) + (1-u) G(y).
end{align}
The question is, what is the most general condition on $f$ such that this requirement is satisfied?
It is easy to verify that for special examples such that $f(t) = log t$ or $f(t) = t$, it is true.
Conjecture: it is true whenever $f$ is a concave function and $f'(1/2) neq 0$.
convex-analysis duality-theorems convex-geometry convex-hulls
asked Dec 23 '18 at 21:09
user401582user401582
686
$endgroup$
add a comment |
$begingroup$
Suppose we have a univariate function $f(t), tin [0,1]$. We define
begin{align}
G(p) = sup_{tin [0,1]} p f(t) + (1-p) f(1-t), text{ for } pin [0,1].
end{align}
Clearly $G(p)$ is a convex function in $p$ since it is the supremum of a family of convex functions in $p$. However, we want a stronger requirement: we would like to make sure that $G(p)$ is strictly convex at point $p = 1/2$. In other words, we want to guarantee that for any $x,yin [0,1], x neq 1/2, yneq 1/2$, any $uin (0,1)$ such that $u x + (1-u) y = 1/2$, we have
begin{align}
G(1/2) < u G(x) + (1-u) G(y).
end{align}
The question is, what is the most general condition on $f$ such that this requirement is satisfied?
It is easy to verify that for special examples such that $f(t) = log t$ or $f(t) = t$, it is true.
Conjecture: it is true whenever $f$ is a concave function and $f'(1/2) neq 0$.
convex-analysis duality-theorems convex-geometry convex-hulls
asked Dec 23 '18 at 21:09
user401582user401582
686
$endgroup$
$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48
add a comment |
$begingroup$
Suppose we have a univariate function $f(t), tin [0,1]$. We define
begin{align}
G(p) = sup_{tin [0,1]} p f(t) + (1-p) f(1-t), text{ for } pin [0,1].
end{align}
Clearly $G(p)$ is a convex function in $p$ since it is the supremum of a family of convex functions in $p$. However, we want a stronger requirement: we would like to make sure that $G(p)$ is strictly convex at point $p = 1/2$. In other words, we want to guarantee that for any $x,yin [0,1], x neq 1/2, yneq 1/2$, any $uin (0,1)$ such that $u x + (1-u) y = 1/2$, we have
begin{align}
G(1/2) < u G(x) + (1-u) G(y).
end{align}
The question is, what is the most general condition on $f$ such that this requirement is satisfied?
It is easy to verify that for special examples such that $f(t) = log t$ or $f(t) = t$, it is true.
Conjecture: it is true whenever $f$ is a concave function and $f'(1/2) neq 0$.
convex-analysis duality-theorems convex-geometry convex-hulls
asked Dec 23 '18 at 21:09
user401582user401582
686
$endgroup$
Suppose we have a univariate function $f(t), tin [0,1]$. We define
begin{align}
G(p) = sup_{tin [0,1]} p f(t) + (1-p) f(1-t), text{ for } pin [0,1].
end{align}
Clearly $G(p)$ is a convex function in $p$ since it is the supremum of a family of convex functions in $p$. However, we want a stronger requirement: we would like to make sure that $G(p)$ is strictly convex at point $p = 1/2$. In other words, we want to guarantee that for any $x,yin [0,1], x neq 1/2, yneq 1/2$, any $uin (0,1)$ such that $u x + (1-u) y = 1/2$, we have
begin{align}
G(1/2) < u G(x) + (1-u) G(y).
end{align}
The question is, what is the most general condition on $f$ such that this requirement is satisfied?
It is easy to verify that for special examples such that $f(t) = log t$ or $f(t) = t$, it is true.
Conjecture: it is true whenever $f$ is a concave function and $f'(1/2) neq 0$.
convex-analysis duality-theorems convex-geometry convex-hulls
convex-analysis duality-theorems convex-geometry convex-hulls
asked Dec 23 '18 at 21:09
user401582user401582
686
asked Dec 23 '18 at 21:09
user401582user401582
686
edited Jan 1 at 19:22
user401582
asked Dec 23 '18 at 21:09
user401582user401582
686
asked Dec 23 '18 at 21:09
user401582user401582
686
asked Dec 23 '18 at 21:09
user401582user401582
686
686
$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48
add a comment |
$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48
$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48
$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48
add a comment |
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$begingroup$
what about $x=y=0.5$?
$endgroup$
– LinAlg
Dec 24 '18 at 2:48