"Convexity" of a family of distributions

Question

Let $\alpha\in(0,1)$. Consider a family of CDFs $\mathcal{X}$ that contains every CDF $X$ defined on $[0,\infty)$ with increasing hazard rate (IHR) which satisfies $$\mathbb{P}[x>y]=\alpha,$$ where $x\sim X$ and $y\sim \textrm{Exponential}(1)$ are independent. I want to show that $\mathcal{X}$ is "convex", in some sense. A mixture of two members of $\mathcal{X}$ satisfies the above equality, but may not have IHR. A convolution of two members of $\mathcal{X}$ satisfies the IHR property but not necessarily the above equation. Are there "natural" addition and scalar multiplication operators defined on $\mathcal{X}$ under which this set is convex?

Answer 1

In fact $\mathcal{X}$ is convex if scalar multiplication is defined as speeding up time and addition is defined as convolution.