On the convexity of a particular form of integrals

Question

EDIT: I made some critical corrections below.

Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, real-valued function. I would like to decide on the convexity of the function $f\colon\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$, given by $$ f(\mathbf{a},b)=\int_{\Omega}\! (\mathbf{a}\cdot\mathbf{x}+b)g(\mathbf{x}) \,\mathrm{d}\mathbf{x}, $$ where $\mathbf{x}\in\mathbb{R}^n$, $b\in\mathbb{R}$, and $\Omega$ is the half-space defined by $\mathcal{H}$ as $\Omega=\{\mathbf{x}\in\mathbb{R}^n\rvert\mathbf{w}\cdot\mathbf{x}+c\geq0\}$.

What I have thought so far is to express the integral as a sum (if such a thing is feasible) and use the property of the affine (and consequently convex) quantity $(\mathbf{a}\cdot\mathbf{x}+b)\cdot k$, where $k\in\mathbb{R}$. Albeit, I am not sure at all this is going to work. In addition, I am not sure in what direction should I work to...

Is there any appropriate argument I could use instead?

Answer 1

Not sure if you can use this, but a basic property of convex functions is that nonnegative combinations of them are convex. This extends to nonnegative integrals as well. If $h(x,t)$ is convex in $x$ and $g$ is nonnegative over the domain $\Omega$, and $$ f(x) = \int_\Omega h(x,t)g(t)dt $$ then $f$ is convex.

This is a consequence of the definition of convexity and some basic properties of integrals:

$$ \begin{aligned} f(\theta x + (1-\theta) y) &= \int_\Omega h(\theta x + (1-\theta) y,t)g(t)dt \\ &\le \int_\Omega (\theta h(x,t) + (1-\theta) h(y,t))g(t)dt \\ &= \theta \int_\Omega h(x,t) g(t)dt + (1-\theta) \int_\Omega h(y,t)g(t)dt \\ &= \theta f(x) + (1-\theta) f(y) \end{aligned} $$

Conversely, if you can express $f$ in this form with a nonconvex $h$, then you might be able to use it to find a example where $f$ is nonconvex.

In your specific case, it seems, the function is not only convex, but affine, which you can see by factoring: $$ f(\mathbf{a},b)=\mathbf{a} \cdot \left(\int_{\Omega}\! \mathbf{x}g(\mathbf{x}) \,\mathrm{d}\mathbf{x}\right) + b \left(\int_{\Omega}\! g(\mathbf{x}) \,\mathrm{d}\mathbf{x}\right) $$

Answer 2

Let me make my question more general and clear, maybe giving an answer, thanks to @p.s, of course!

Let $f\colon K\subseteq\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$, given by $$ f(\mathbf{a},b)=\int_{\Omega}\! h(\mathbf{a},b,\mathbf{x})g(\mathbf{x}) \,\mathrm{d}\mathbf{x}, $$ where $\Omega=\{\mathbf{x}\in\mathbb{R}^n\rvert\mathbf{w}\cdot\mathbf{x}+c\geq0\}$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$ be a non-negative function.

If $h$ is convex with respect to $\mathbf{a}$, $b$ over the domain $\Omega$, then $f$ is convex in $K$.

Proof Let $\lambda\in[0,1]$ and $(\mathbf{a}_1,b_1),(\mathbf{a}_2,b_2)\in K$. Then, $$ \begin{aligned} f(\lambda\mathbf{a}_1+(1-\lambda)\mathbf{a}_2,\lambda b_1+(1-\lambda)b_2) &= \int_\Omega\!h(\lambda\mathbf{a}_1+(1-\lambda)\mathbf{a}_2,\lambda b_1+(1-\lambda)b_2,\mathbf{x})g(\mathbf{x})\,\mathrm{d}\mathbf{x} \\ &\le \int_\Omega\!\left[\lambda h(\mathbf{a}_1,b_1,\mathbf{x}) + (1-\lambda) h(\mathbf{a}_2,b_2,\mathbf{x})\right]g(\mathbf{x})\,\mathrm{d}\mathbf{x} \\ &= \lambda \int_\Omega h(\mathbf{a}_1,b_1,\mathbf{x}) g(\mathbf{x})d\mathbf{x} + (1-\lambda) \int_\Omega\!h(\mathbf{a}_2,b_2,\mathbf{x})g(\mathbf{x})\,\mathrm{d}\mathbf{x} \\ &= \lambda f(\mathbf{a}_1,b_1) + (1-\lambda) f(\mathbf{a}_2,b_2). \end{aligned} $$

So, I think this can be applied to my problem, as the function $h(\mathbf{a},b,\mathbf{x})=\mathbf{a}\cdot\mathbf{x}+b$ is convex (actually it is affine with respect to $\mathbf{a}$, $b$) with respect to $\mathbf{a}$, $b$.

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A Counterexample?

Let $h(\mathbf{a},b,\mathbf{x})=\mathbf{a}\cdot\mathbf{x}+b+1$, which is affine w.r.t. $\mathbf{a},b$ over the domain $\Omega=\{\mathbf{x}\in\mathbb{R}^n \mid \mathbf{a}\cdot\mathbf{x}+b+1 \geq 0\}$. Also, let $g$ be the probability density function of an $n$-dimensional Gaussian with mean $\mu\in\mathbb{R}^n$ and covariance matrix $\Sigma\in\mathbb{S}_{++}^n$ (symmetric, positive-definite, $n\times n$). Then, the function $$ f(\mathbf{a},b)=\int_{\Omega}\! h(\mathbf{a},b,\mathbf{x})g(\mathbf{x}) \,\mathrm{d}\mathbf{x} $$ is argued to be convex from the above "Theorem", but it is evaluated as follows

$$ f(\mathbf{a},b) = \frac{\mathbf{a}\cdot\mu+b+1}{2} \left[ 1 - \operatorname{erf}\left(-\frac{\mathbf{a}\cdot\mu+b+1}{\sqrt{2\mathbf{a}^T\Sigma\mathbf{a}}}\right) \right] + \frac{\sqrt{\mathbf{a}^T\Sigma\mathbf{a}}}{2\pi} \exp\left( -\frac{(\mathbf{a}\cdot\mu+b+1)^2}{2\mathbf{a}^T\Sigma\mathbf{a}}\right), $$

which obviously is not convex in $(\mathbf{a},b)$.

Where is the mistake?