Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.