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There is no such "nice" $C$ (other than $\Omega$ and $\varnothing$) in dimensions other than $1$. Suppose otherwise, that a non-empty, smooth, open set $C$ in $\mathcal{A}(\Omega)$ exists. If $C \ne …

The answer to the new question is still no. Let us identify $\mathbb{R}^2$ with $\mathbb{C}$. Consider $\varphi(z) = \log |\sin z|$ for $z \in \mathbb{C}$. This is clearly a subharmonic function. Let …

Consider $$ f(x, y) = \max \biggl\{ \frac{3 x}{2^n} - \frac{1}{4^n} + \frac{(-1)^n y}{n + 1} : n = 0, 1, 2, \ldots \biggr\} . $$ Then $f$ is clearly convex, and $$ \partial_y f(x, 0) = \frac{(-1)^n}{n …

Let $$\begin{aligned} f(x,y) & = x^4 - x^2 y^2 + y^4 \\ & = \tfrac{1}{2} x^4 + \tfrac{1}{2} y^4 + \tfrac{1}{2} (x^2 - y^2)^2 . \end{aligned}$$ Then $f$ is a strictly positive (except at the origin, of …

Theorem 2(b) in [1] is equivalent to log-convexity of $K_\nu$ for every $\nu$. This is said to be "well-known", and three references are given. [1] Árpád Baricz, Saminathan Ponnusamy, Matti Vuorinen, …