# Extending function from hyperplane segment to ball, estimating integral by integral on manifold

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.