# Intuition behind One-sided Lipschitz

In my current lecture Numerical Analysis of Ordinary Differential Equations we introduced the concept of One-sided Lipschitz functions.

A function $$f: D \rightarrow \mathbb{C}^d$$ satisfies a one-sided Lipschitz condition on it's domain $$D \subseteq \mathbb{R} \times \mathbb{C}^d$$ if there exists $$C \in \mathbb{R}$$, such that the inequality

$$\Re((f(t,x) - f(t,y))^t(x-y)) \leq C|x-y|^2$$

holds for all $$(t,x),(t,y) \in D$$.

I always visualised the Lipschitz condition using this cone-intuition, but can't seem to find an analogue for the one-sided Lipschitz condition. Is there a visual interpretation of the one-sided version that, for example helps me understand why $$f(x) = e^{-x}$$ satisfies a one-sided Lipschitz condition?

As you have already mentioned, Lipschitz condition is sometimes illustrated by a cone condition:

Note that the Lipschitz constant $$C$$ is related to the (white) cone angle, i.e. if $$\theta$$ is the angle between two generating lines of the cone (which we call $$L_1$$ and $$L_2$$) then $$C=\cot\frac{\theta}2$$ The key to understand or visualize one-sided Lipschitz condition is the $$\frac{\theta}2$$ term. If a function is not Lipschitz at all, then there is no $$C$$ or in other words, $$C$$ tends to infinity and thus $$\frac{\theta}2$$ or $$\theta$$ are zero. This means that the cone vanishes.

Let $$\alpha_1$$ be the angle between $$L_1$$ and the vertical line. Also let $$\alpha_2=\theta-\alpha_1$$. For a Lipschits function, these are both equal to $$\frac{\theta}2$$. But if a function is Lipschitz only in one direction, it would mean either of $$\cot\alpha_1$$ or $$\cot\alpha_2$$ is undefined. So either $$\alpha_1=0$$ or $$\alpha_2=0$$. In other words, the cone is oblique.

For example, $$f(x)=e^{-x}$$ is one-sided Lipschitz with $$C=0$$. You can easily verify that in this case $$\alpha_1=90^\circ$$ and $$\alpha_2=0$$.

Edit: How to mathematically formulate this property?

For real functions of a single variable, the relationship between the intuition and mathematical formulation is straightforward. In this case, if a function is Lipschitz on a domain $$D\subset\mathbb R$$ then $$|f(x_1)-f(x_2)|\le C|x_1-x_2|,\quad \{x_1,x_2\}\subset D$$ And if it is one-sided Lipschitz, there would be no absolute values in this inequality. For example, if $$f$$ is Lipschitz on the right side then for every real values of $$x_1$$ and $$x_2$$ on its domain, there is $$C\ge 0$$ such that $$f(x_2)-f(x_1)\le C(x_2-x_1)$$ which can be re-written as $$\left(f(x_2)-f(x_1)\right)(x_2-x_1)\le C|x_2-x_1|^2\tag{*}$$ Now for multi-variable complex functions, the $$(*)$$ condition can be generalized to the form you mentioned: $$\Re\{\left(f(x_2) - f(x_1)\right)^T(x_2-x_1)\} \leq C||x_2-x_1||^2$$

• Thank you. Is there also a way to justify this intuition using the definition of the one sided Lipschitz condition? – Herickson Jul 12 '19 at 11:20
• @Herickson see the edited answer – polfosol Jul 13 '19 at 13:16