# Convexity of p power of the q norm (0<p<1, q>1)

I encountered a minimization problem involving the following function:

$$f(\mathbf{x})=\|\mathbf{x}\|_q^p$$

Here, $$q>1$$ and $$0. Naturally, each entry of $$\mathbf{x}$$ is greater than $$0$$.

I would like to determine the convexity (or concavity) of $$f$$, but I'm uncertain about how to do it. In my previous experience, I could demonstrate convexity by checking the definition or using rules of convexity, such as the convexity of the sum of two functions or certain composition rules. Or, I could prove the positive semi-definiteness of the Hessian. However, I haven't been able to apply any of these techniques to analyze $$f$$.

Assuming $$q > 1, 0 , as per the question:

If $$x$$ is a nonnegative scalar, $$f(\mathbf{x})=\|\mathbf{x}\|_q^p$$ is concave, not convex.

For non-scalar $$x$$, $$f(\mathbf{x})=\|\mathbf{x}\|_q^p$$ is neither convex nor concave. For instance, for p = 0.5, q = 2, in 2D at $$x = [1, 1]^T$$, the Hessian of $$f(x)$$ has eigenvalues of 0.297 and -0.149; so the Hessian is indefinite there, and therefore $$f(x)$$ is neither convex nor concave.

• What value of $p$ are you using in your example? For $p=1$ (granted just outside the specified scope) and $q=2$, $f$ would be the euclidean 2 norm.
– prubin
Dec 8, 2023 at 18:54
• @Mark L. Stone, would you please, where can we find a material to learn about the norms and their specifications? (I know there are some books, but I appreciate if it has been dense with focus on only norms). Dec 8, 2023 at 19:22
• @ptubin Sorry. I forgot to include the value of p used in my example, which is 0.5. I just edited my answer to include that. It wasn't hard fr me to find a "counterexample". The set of values in my answer was the first ones I tried. I come up with such counterexamples to convexity or concavity very frequently over at ask.cvxr.com . Dec 8, 2023 at 21:06
• @A.Omidi Read and solve some exercises in at least the first 4 or 5 chapters of "Convex Optimization" by Boyd and Vandenberghe web.stanford.edu/~boyd/cvxbook to learn something about convex optimization, and determining convexity and concavity. You can pick up something about norms from Appendix A and supplement that with en.wikipedia.org/wiki/Norm_(mathematics) . Dec 8, 2023 at 21:11
• @prubin, with $q = 2$, and $x = [1,1]^T, p = 1-10^{-9}$ is not convex (nor concave). With other values of $x$, perhaps $p = 1 - \epsilon$ is sufficient for non-convexity. Even though q = 2 is not on the edge of convexity for a norm (which is at q = 1), there's no room to spare for allowing $0 < p < 1$ Dec 8, 2023 at 21:27