2 typo pointed out by fedja, thanks; deleted 17 characters in body

First a little background. Mircowaves do not heat uniformly. To help overcome this, your 'food' is rotated, however this is not usually sufficient to produce a totally uniform heating. Informally, the question is when can we find a way of moving our 'food' in order to heat it uniformly throughout?

Let $f : \mathbb{R}^n \to R$ be our heat function. Let $I^n = [-0.5,0.5] \times \cdots \times [-0.5,0.5]$ be the unit n-dimensional cube centered at the origin, this will be our 'food'. Let $\gamma : [0,1] \to \mathbb{R}^n \times SO(n)$ be a map specifying a path along which to translate and rotate $I^n$. If $x \in I^n$ then let $h(x)$ denote the total 'heat absorbed' by $x$ as it travels along $\gamma$. Note if $\gamma(t) = (\gamma_1(t), \gamma_2(t))$ then $h(x) = \int_0^1 \gamma_2(t)(x) f(\gamma_2(t)(x) + \gamma_1(t) gamma_1(t)) dt$.

We will call a curve $\gamma$ uniformly heated iff $\forall x,y \in I^n$, $h(x) = h(y)$.

How sufficiently 'nice' must our heat function $f$ be in order to guarantee that there exists a uniformly heated curve? Do these requirements change if we consider a different 'food' to heat? For example, if we heat $I^m \times 0^{n-m}$ in $\mathbb{R}^n$.

Note that in $\mathbb{R}^1$, as $SO(1) = 1$, if $f$ is a non-constant and strictly monotonic function then there cannot exist any uniformly heated curves as (assuming wlog $f$ is increasing) $h(-0.5) < h(0.5)$.

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# Microwaving Cubes

First a little background. Mircowaves do not heat uniformly. To help overcome this, your 'food' is rotated, however this is not usually sufficient to produce a totally uniform heating. Informally, the question is when can we find a way of moving our 'food' in order to heat it uniformly throughout?

Let $f : \mathbb{R}^n \to R$ be our heat function. Let $I^n = [-0.5,0.5] \times \cdots \times [-0.5,0.5]$ be the unit n-dimensional cube centered at the origin, this will be our 'food'. Let $\gamma : [0,1] \to \mathbb{R}^n \times SO(n)$ be a map specifying a path along which to translate and rotate $I^n$. If $x \in I^n$ then let $h(x)$ denote the total 'heat absorbed' by $x$ as it travels along $\gamma$. Note if $\gamma(t) = (\gamma_1(t), \gamma_2(t))$ then $h(x) = \int_0^1 \gamma_2(t)(x) + \gamma_1(t) dt$.

We will call a curve $\gamma$ uniformly heated iff $\forall x,y \in I^n$, $h(x) = h(y)$.

How sufficiently 'nice' must our heat function $f$ be in order to guarantee that there exists a uniformly heated curve? Do these requirements change if we consider a different 'food' to heat? For example, if we heat $I^m \times 0^{n-m}$ in $\mathbb{R}^n$.

Note that in $\mathbb{R}^1$, as $SO(1) = 1$, if $f$ is a non-constant and strictly monotonic function then there cannot exist any uniformly heated curves as (assuming wlog $f$ is increasing) $h(-0.5) < h(0.5)$.