The generalized-cone extremal conjecture for concave functions

Let $\mathcal K^n$ denote the class of convex bodies in $\mathbb R^n$. Let $C\in\mathcal K^n$, let $f:C\rightarrow[0,\infty)$ be concave, and let $\phi:[0,\infty)\rightarrow[0,\infty)$ be convex with $\phi(0)=0$. Write $x_{C,f}$ for the center associated with $C$ and $f$. A generalized cone is a convex body obtained as the convex hull of a base and a point outside the affine hull of that base. Then

Generalized-cone extremal conjecture.

$$\frac{1}{|C|}\int_C\phi(f(x))\,dx \leq n\int_0^1\phi\left(f(x_{C,f})\frac{2^{1/n}t}{2^{1/n}-1}\right)(1-t)^{n-1}\,dt.$$

Moreover, if $\phi$ is strictly convex, equality holds if and only if $C$ is a generalized cone and $f$ is an affine function which becomes zero at the base of $C$.

The conjecture proposes that the maximum in the general inequality is attained by a generalized cone and an affine function vanishing on its base. It is motivated by the corresponding equality cases established in the paper for several two- and three-dimensional instances; the general-dimensional assertion remains open in the supplied text.