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Let $f : \mathbb R^n \rightarrow \mathbb R$ be continuous and Lebesgue integrable. Let $m^n$ denote $n$-dimensional Lebesgue measure. Suppose that for each $n-1$ dimensional hyperplane $H$ in $\mathbb R^n$, we have

$$\int_H f_H dm^{n-1} = 0,$$

where $f_H$ is the restiction of $f$ to $H$, and $H$ is identified with $\mathbb R^{n-1}$ via an isometry. Must $f$ be identically 0?

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