Return to Question

We are given a triangle $T$ on a plane $P$, with sides $A$, $B$ and $C$, whose lengths are $a$, $b$ and $c$ respectively, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the horizontal and vertical straight lines cutting $T$. Note that a.s. there is $1$ and only $1$ uncut side of $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$ on $P$, where the expectation is taken over the random selection of $L$?

We are given a triangle $T$ on a plane $P$, with sidelengths $a$, $b$ and $c$, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the horizontal and vertical straight lines cutting $T$. Note that a.s. there is $1$ and only $1$ uncut side of $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$ on $P$, where the expectation is taken over the random selection of $L$?

We are given a triangle $T$ on a plane $P$, with sides $A$, $B$ and $C$, whose lengths are $a$, $b$ and $c$ respectively, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the horizontal and vertical straight lines cutting $T$. Note that a.s. there is $1$ and only $1$ uncut side of $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$, i.e., over all orientations of $T$ on the plane and all possible choices of $a$, $b$ and $c$ , where the expectation is taken over the random selection of $L$?

We are given a triangle $T$ on a plane $P$, with sides $A$, $B$ and $C$, whose lengths are $a$, $b$ and $c$ respectively, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the horizontal and vertical straight lines cutting $T$. Note that a.s. there is $1$ and only $1$ uncut side of $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$ on $P$, where the expectation is taken over the random selection of $L$?

We are given a triangle $T$ on a plane $P$, with sides $A$, $B$ and $C$, whose lengths are $a$, $b$ and $c$ respectively, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected as follows. With probability $\tfrac12$, $L$ is selected uniformly at random from the set of all the horizontal straight lines cutting $T$, and with probability $\tfrac12$, $L$ is selected uniformly at random from the set of all vertical straight lines cutting $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$, i.e., over all orientations of $T$ on the plane and all possible choices of $a$, $b$ and $c$ , where the expectation is taken over the random selection of $L$?

We are given a triangle $T$ on a plane $P$, with sides $A$, $B$ and $C$, whose lengths are $a$, $b$ and $c$ respectively, where $c \ge b \ge a > 0$. A straight line $L$ on $P$ is selected uniformly at random from the set of all the horizontal and vertical straight lines cutting $T$. Note that a.s. there is $1$ and only $1$ uncut side of $T$.

Question: What maximum expected length (as a function of $a$, $b$ and $c$) of the uncut side of $T$ over all possible triangles $T$, i.e., over all orientations of $T$ on the plane and all possible choices of $a$, $b$ and $c$ , where the expectation is taken over the random selection of $L$?