# Editing Open Problems:70

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A tolerant $L_1$ property tester has to distinguish functions $f$ that are $\epsilon_1$-close to a property $P$ ($\inf_{g \in P} \operatorname{dist}_1(f,g) \le \epsilon_1$) from those that are $\epsilon_2$-far ($\inf_{g \in P} \operatorname{dist}_1(f,g) \ge \epsilon_2$). | A tolerant $L_1$ property tester has to distinguish functions $f$ that are $\epsilon_1$-close to a property $P$ ($\inf_{g \in P} \operatorname{dist}_1(f,g) \le \epsilon_1$) from those that are $\epsilon_2$-far ($\inf_{g \in P} \operatorname{dist}_1(f,g) \ge \epsilon_2$). | ||

β | * '''Problem 1:''' {{cite|BermanRY-14}} describe | + | * '''Problem 1:''' {{cite|BermanRY-14}} describe an $L_1$-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 2:''' Obtain a tolerant $L_1$ tester for monotonicity for $d\geq 3$. (There exist testers, albeit maybe non-optimal, in the case $d=1$ or $d=2$, from {{cite|BermanRY-14}}; nothing non-trivial is known for higher dimensions.) | * '''Problem 2:''' Obtain a tolerant $L_1$ tester for monotonicity for $d\geq 3$. (There exist testers, albeit maybe non-optimal, in the case $d=1$ or $d=2$, from {{cite|BermanRY-14}}; nothing non-trivial is known for higher dimensions.) | ||

'''Note:''' Slides describing the setting and open problems can be found on [http://grigory.us/#lp-testing Grigory's webpage]. Slides of a longer talk are available [http://grigory.us/files/talks/BRY-STOC14.pdf here]. | '''Note:''' Slides describing the setting and open problems can be found on [http://grigory.us/#lp-testing Grigory's webpage]. Slides of a longer talk are available [http://grigory.us/files/talks/BRY-STOC14.pdf here]. |