# Editing Open Problems:70

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Extending the usual setting of property testing to functions $f\colon\{1,\dots,n\}^d\to[0,1]$, Berman et al. {{cite|BermanRY-14}} study property testing with regard to $L_p$ distances between functions. Namely, these distances are defined as $\operatorname{dist}_p(f,g) = \frac{\lVert f-g\rVert_p}{\lVert \mathbf{1}\rVert_p}$ (for $p > 0$), so that for instance $\operatorname{dist}_1(f,g) = \frac{\lVert f-g\rVert_1}{n^d}$ (and if the functions are Boolean, we get back the Hamming distance). | Extending the usual setting of property testing to functions $f\colon\{1,\dots,n\}^d\to[0,1]$, Berman et al. {{cite|BermanRY-14}} study property testing with regard to $L_p$ distances between functions. Namely, these distances are defined as $\operatorname{dist}_p(f,g) = \frac{\lVert f-g\rVert_p}{\lVert \mathbf{1}\rVert_p}$ (for $p > 0$), so that for instance $\operatorname{dist}_1(f,g) = \frac{\lVert f-g\rVert_1}{n^d}$ (and if the functions are Boolean, we get back the Hamming distance). | ||

β | Let $P$ be a class of functions (e.g. monotone, convex, Lipschitz, etc.) A non-tolerant $L_1$ property tester has to distinguish functions $f$ that have a property $P$ from those that are $\epsilon$-far, i.e. $\inf_{g \in P} | + | Let $P$ be a class of functions (e.g. monotone, convex, Lipschitz, etc.) A non-tolerant $L_1$ property tester has to distinguish functions $f$ that have a property $P$ from those that are $\epsilon$-far, i.e. $\inf_{g \in P} dist_1(f,g) \ge \epsilon$. |

β | A tolerant $L_1$ property tester has to distinguish functions $f$ that are $\epsilon_1$-close to a property $P$ ($\inf_{g \in P} | + | A tolerant $L_1$ property tester has to distinguish functions $f$ that are $\epsilon_1$-close to a property $P$ ($\inf_{g \in P} dist_1(f,g) \le \epsilon_1$) from those that are $\epsilon_2$-far ($\inf_{g \in P} dist_1(f,g) \ge \epsilon_2$). |

* '''Problem 1:''' {{cite|BermanRY-14}} describe a non-tolerant $L_1$-tester for convexity whose query complexity, $O(\frac{1}{\varepsilon^{d/2}}+\frac{1}{\varepsilon})$, grows exponentially with the dimension $d$. Is this exponential dependence necessary, or is there a tester with query complexity $O(\frac{1}{\varepsilon^{o(d)}})$? | * '''Problem 1:''' {{cite|BermanRY-14}} describe a non-tolerant $L_1$-tester for convexity whose query complexity, $O(\frac{1}{\varepsilon^{d/2}}+\frac{1}{\varepsilon})$, grows exponentially with the dimension $d$. Is this exponential dependence necessary, or is there a tester with query complexity $O(\frac{1}{\varepsilon^{o(d)}})$? |