Question
Consider a function $f:\mathbb R \to \mathbb R$. Are the following conditions equivalent?
- Q1. The function $f$ is continuously differentiable and it second derivative $f^{(2)}$ exists at all but countably many points and is increasing on its domain.
- Q2. For any $a<b<c$ and the quadratic function $q(x)$ interpolating the points $(a,f(a)),(b,f(b)),(c,f(c))$, the graph of the restriction $q|_{[b,c]}$ is contained in the epigraph $\mathop{epi}(f)$.
- Q3. For any $a<b<c<d$ and a quadratic function $q(x)$ if both of the points $(a,q(a))$ and $(c,q(c))$ are contained in $\mathop{epi}(f)$ whilst both of the points $(b,q(b))$ and $(d,q(d))\}$ are outside of the epigraph then the graph of $q|_{(-\infty,a]}$ is contained in $\mathop{epi}(f)$.
And is the following condition necessary condition for Q1-Q3?
- Q0. The third derivative $f^{(3)}$ exists almost everywhere and is positive on its domain.
Notation:
Positivness and monotonicity are considered in the weak sense, i.e. using weak inequalities "$\geq$".
Almost everywhere refers to a property being satisfied everywhere except for a set of zero Lebesgue measure.
Graph of the restriction $q|_{[b,c]}$ is the set $\mathop{graph}(q|_{[b,c]})=\{(x,q(x))\in \mathbb R^2:x\in [b,c]\}.$
Epigraph of $f$ is the set $\mathop{epi}(f) = \{(x,y)\in \mathbb R^2:y \geq f(x)\}$.
Motivation:
Condition Q1 implies Q0 due to the fact that a monotone function is differentiable almost everywhere.
Condition Q2 abstracts from Q0 and Q1 by not referring to any of the function's derivatives explicitly.
Condition Q3 is an attempt to formulate Q2 by referring solely to the epigraph of $f$ and not its boundary – the graph of $f$ – similarly as convexity of a set $S$ is usually defined by requiring that any segment connecting any two points in $S$ is contained in $S$ (instead of requiring this property only for any two points on the boundary of $S$). Notice that the graph of a quadratic function can be characterized as a parabola with a vertical axis of symmetry.
Observations:
- If $f$ is $3$ times differentiable then Q0 is equivalent to Q2: Geometric characterization of the $n$-th derivative of $f$ being positive (convexity for $n=2$).
- The implication Q1 $\Longrightarrow$ Q2 is discussed in A higher-order analogy to that if $f'$ is increasing on $\Bbb R$ except for countably many points, then $f$ is convex and its reverse in A higher-order analogy to that every convex function is differentiable at all but countably many points.
Analogy
Conditions Q0-Q3 were motivated by their analogy to corresponding conditions describing monotonicity and convexity.
Convex Functions:
- C0. The second derivative $f''$ exists almost everywhere and is positive on its domain.
- C1. The function $f$ is continuous and its derivative $f'$ exists at all but countably many points and is increasing on its domain.
- C2. For any $a<b$ and the linear function $l(x)$ interpolating the points $(a,f(a)),(b,f(b))$, the graph of $l|_{[a,b]}$ is contained in $\mathop{epi}(f)$.
- C3. For any $a<b$ and a linear function $l(x)$ if both of the points $(a,l(a)),(b,l(b))$ are in $\mathop{epi}(f)$ then the graph of $l|_{[a,b]}$ is contained in $\mathop{epi}(f)$.
Note that Condition C3 is equivalent to $\mathop{epi}(f)$ being a convex set.
Condition C0 is a necessary but not sufficient condition for convexity: https://math.stackexchange.com/a/3425608/1134951
Increasing Functions:
- I0. The derivative $f'$ exists almost everywhere and is positive on its domain.
- I1. The function $f$ is increasing.
- I2. For any $a\in \mathbb R$ and the constant function $m(x)$ interpolating the point $(a,f(a))$, the graph of $m|_{(-\infty,a]}$ is contained in $\mathop{epi}(f)$.
- I3. For any $a\in \mathbb R$ and a constant function $m(x)$ if the point $(a,m(a))$ is in $\mathop{epi}(f)$ then the graph of $m|_{(-\infty,a]}$ is contained in $\mathop{epi}(f)$.
Note that Condition I3 is equivalent to $\mathop{epi}(f)$ being star-shaped with the vantage point $(-\infty,0)$ at the infinity. In other terms $\mathop{epi}(f) = \mathop{epi}(f) + (-\infty,0]\times\{0\}$. (The vantage point at infinity represents a direction, since the direction is parallel to the $x$-axis, it is possible to informally describe the vantage point as $(-\infty,0)$ instead of representing it in the homogenous coordinates as $(-1,0,0)$.)
Let me recall that the question was whether Conditions Q1-Q3 are equivalent and if they imply Condition Q0.
