A Lipschitz Estimate Between the Maximisers of Two Linear Perturbations of a Semiconvex Function

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure; $2$ denotes $1+1$, which satisfies $0<2$ and so has a multiplicative inverse by claim 8 of Elementary Order Arithmetic in an Ordered Field, and $\tfrac{a}{2}$ denotes the product of $a$ with that inverse. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $y-x$ and dot product $p\cdot x$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ with $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Closed balls $\bar{B}_{d_{E}}$ are those of Closed Ball in a Metric Space.

Let $U\subseteq\mathbb{R}^{n}$ be convex, let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$, and let $\varphi:U\to\mathbb{R}$ be semiconvex on $U$ with constant $\lambda$. Let $\hat{x}\in\mathbb{R}^{n}$ and $r\in\mathbb{R}$ satisfy $0<r$ and $\bar{B}_{d_{E}}(\hat{x},r)\subseteq U$, and write $\bar{B}=\bar{B}_{d_{E}}(\hat{x},r)$.

Let $p,p'\in\mathbb{R}^{n}$, let $x\in\bar{B}_{d_{E}}\bigl(\hat{x},\tfrac{r}{2}\bigr)$ and let $x'\in\bar{B}$ satisfy

$$\varphi(y)+p\cdot y\le\varphi(x)+p\cdot x\quad\text{and}\quad \varphi(y)+p'\cdot y\le\varphi(x')+p'\cdot x'\qquad\text{for every }y\in\bar{B},$$

and suppose in addition that

$$\lVert p-p'\rVert\le\frac{\lambda r}{2}.$$

Then the following hold.

1. (Monotonicity estimate) (i)

$$\lVert p-p'\rVert^{2}\le 2\lambda\,(p-p')\cdot(x-x').$$

2. (Lipschitz estimate) (ii)

$$\lVert p-p'\rVert\le 2\lambda\,\lVert x-x'\rVert .$$

In particular, if $x=x'$ then $p=p'$.