On the convexity of certain set of random vectors Let ${\cal X}$ be the set of pairs of random variables $(X,Y)$ with finite expectations. For constant $c\in[0,1]$, define set
$$
\{(X,Y)\in{\cal X}:\exists a\geq 0, \, b\geq 0 \text{ such that } E[\max(aX+1,bY+1,0)] \leq c\}.
$$
The question is whether this set is convex.
If this helps, the convexity of the corresponding set of random variables
$$
\{X:\exists a\geq 0 \text{ such that } E[\max(aX+1,0)] \leq c\}
$$
is known.
 A: $\renewcommand{\P}{\operatorname{\mathsf P}}\newcommand\E{\operatorname{\mathsf{E}}}$The answer is no. Indeed, suppose that $(X_1,Y_1)$ and $(X_2,Y_2)$ are pairs of random variables (r.v.'s) such that
\begin{equation}
    \P\big((X_1,Y_1)=P_{1,i},\,(X_2,Y_2)=P_{2,k}\big)=p_{i,k},
\end{equation}
for $i,k=1,2$, where
\begin{equation}
    (P_{1, 1}, P_{1, 2}, P_{2, 1}, P_{2, 2})
    :=\frac1{100}\,\big((46, 157), (-168, -183), (-144, -12), (89, -245)\big), 
\end{equation}
\begin{equation}
    \left(
\begin{array}{cc}
 p_{1,1} & p_{1,2} \\
 p_{2,1} & p_{2,2} \\
\end{array}
\right)
:=\frac1{100}\,
\left(
\begin{array}{cc}
 31 & 11 \\
 35 & 23 \\
\end{array}
\right),
\end{equation}
\begin{equation}
    a_1:=\frac{7850}{4209}, \quad b_1:=\frac{100}{183},\quad 
    a_2:=\frac{25}{36},\quad b_2:=\frac{25}{3}; 
\end{equation}
note that $\sum_{i,k=1}^2 p_{i,k}=1$.
Then
\begin{equation}
    \E\max(a_1X_1+1,b_1Y_1+1,0)=\frac{238}{305}=0.78032\ldots
\end{equation}
and
\begin{equation}
\E\max(a_2X_2+1,b_2Y_2+1,0)=\frac{3961}{7200}=0.55013\ldots,
\end{equation}
so that the random pairs $(X_1,Y_1)$ and $(X_2,Y_2)$ of r.v.'s are in your set,
\begin{equation}
    S(c):=\{(X,Y)\in{\mathcal X}:\exists a\ge0, \, b\ge0 \ \E\max(aX+1,bY+1,0)\le c\},
\end{equation}
for
\begin{equation}
    c:=\frac{238}{305}=0.78032\ldots\in[0,1]. 
\end{equation}
However, the minimum
\begin{equation}
    \min_{a,b\ge0}\E\max\Big(a\frac{X_1+X_2}2+1,b\frac{Y_1+Y_2}2+1,0\Big)
    =\frac{129169}{154050}=0.83848\ldots 
\end{equation}
(attained at $(a,b)=(\frac{200}{79},\frac{40}{39})$)
is $>c$, and therefore the random pair
\begin{equation}
    \frac{(X_1,Y_1)+(X_2,Y_2)}2
\end{equation}
is not in the set $S(c)$.
Thus, the set $S(c)$ is not convex.

Below is the image of a Mathematica notebook verifying the above calculations:


