Guarantee strict convexity at a point












1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    what about $x=y=0.5$?
    $endgroup$
    – LinAlg
    Dec 24 '18 at 2:48
















1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    what about $x=y=0.5$?
    $endgroup$
    – LinAlg
    Dec 24 '18 at 2:48














1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 1 at 19:22







user401582

















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


















  • $begingroup$
    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










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