# Lipschitz continuity result

Hi: I'm reading a proof in a convex optimization book and I don't understand something. The author states the following:

Let $Q$ be a subset of $R^{n}$. We denote by $C_{L}^{k,p}(Q)$, the class of functions with the following properties:

A) any $f \in C_{L}^{k,p}(Q)$ is $k$ times differentiable on $Q$.

B) Its $p$-th derivative is Lipschitz continuous on $Q$ with the constant $L$.

i.e: $\|f^{p}(x) - f^{p}(y)\| \le L\|x - y\|$.

This is fine. But then later, inside the proof, he states the following:

if $f \in C_{L}^{2,1}(R^{n})$, then for any $s \in R^{n}$ and $\alpha > 0$, we have

$$\left\|\left(\int_{0}^{\alpha}f''(x + \tau ~ s)d ~ \tau\right) s\right\| = \|f^{\prime}(x + \alpha s) - f^{\prime}(x) \| \le \alpha L \|s\|\;.$$

I don't understand how this is implied by the definition of Lipschitz continuity defined previously. Thanks a lot.

-

## 1 Answer

Just putting in the definition of $f\in C^{2,1}_L(\mathbb{R}^n)$ gives

$$\|f^{(1)}(x+\alpha s)-f^{(1)}(x)\|\le L\|x+\alpha s -x\|=L\|\alpha s\| =L\alpha \|s\|$$

where I used positive homogeneity of the norm $\|\cdot\|$ in the last equality.

-

Hi Pavel: The book I am reading is titled "Introductory Lectures on Convex Optimization" by Yuri Nesterov. The guy is a god in the field so I'm pretty sure the statement is correct. Hi n.c.: Thank you for your help. I'll print it out and see if I follow it. – mark leeds Jan 5 at 15:06
No need to; I misread the definition. – user53153 Jan 5 at 15:57
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9057154655456543, "perplexity_flag": "head"}