Let $B:=B_R (0) \subset \mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H \subset \mathbb{R}^d$ a hyperplane that satisfies $H \cap B \neq \emptyset$ and $0 \not\in H$. Furthermore, let $f \in C^1 (H)$ with $f \geq 0$.

Consider the function $g : \, B \to \mathbb{R}$ given by \begin{equation*} g (x) = \begin{cases} f(\xi), & \xi = (0,x) \cap H \neq \emptyset \\ 0, & \text{otherwise} \end{cases} \end{equation*}

where $(0,x)$ denotes an open line-segment. Note that $g$ is piecewise constant in $x-$direction. Basicly, $g$ extends $f$ from $H$ to $B$ by assigning $f(\xi)$ along any $x-$direction from $H$ towards the boundary of $B$.

I am interested in deriving an estimate of the form \begin{equation*} \int_B g(x) \, dx \leq c \, R \int_{H \cap B} f(\xi) d\xi , \end{equation*}

where $c$ is a constant only depending on the dimension $d$, but am having trouble proving it (nicely). Here is my approach so far:

  1. Transform integral to spherical coordinates with transformation $\Psi$: \begin{equation*} \int_B g(x) \, dx = \int_{S^{d-1}} \int_{0}^{R} g (\Psi^{-1}(r, s)) \, r^{d-1} \, dr \, ds \end{equation*}

  2. Parametrize $H \cap B$ over $S^{d-1}$: Let $U = \lbrace s \in S^{d-1}: \, (0, 2 R s) \cap H \cap B \neq \emptyset \rbrace$ and $h: \, U \to \mathbb{R}$ a suitable $C^1 (U)$ function, such that $\gamma :\, U \to H$ defines a parametrization of $H$ over $U \subset S^{d-1}$ by $\gamma (s) = \Psi^{-1} (h(s), s)$. $H$ is the graph of $h$, so its first fundamental form satisfies $g^{\gamma} (s) = 1 + \Vert \nabla h (s) \Vert^2 \geq 1$. Thus, \begin{align*} & \phantom{{}={}} \int_{S^{d-1}} \int_{0}^{R} g (\Psi^{-1} (r, s)) \, r^{d-1} \, dr \, ds \leq \int_{S^{d-1}} \frac{1}{d} R^{d} f \left(\Psi^{-1}(h (s), s) \right) \, ds \\ & \leq \frac{1}{d} R \int_{U} f (\Psi^{-1} (h(s),s)) \, \sqrt{g^{\gamma}} \, ds = \frac{1}{d} R \int_{H \cap B} f (\xi) \, d\xi \end{align*}

Any hints or comments are appreciated.


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