EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the graph of $f$ has no supporting hyperplanes with zero gradient (more explicitly, any supporting hyperplane of the function $f(x)+\lambda |x|^2$ at a point $x$ is not parallel to the supporting functional of the function $\lambda |x|^2$ at the same point $x$).
Question. Is it true that each level set $f^{-1}(c)$ has Hausdorff dimension $n-1$? If yes, is it true that the corresponding Hausdorff measure of it is locally finite?
The case when $f$ is convex (i.e. $\lambda =0$) is known to be true since $\{f\leq c\}$ is a convex set, and its boundary is known to have Hausdorff dimension at most $n-1$ of locally finite Hausdorff measure.
