6 Minor edit.

You can use the triangle inequality to solve this by looking at each coordinate separately.

$F(x_0,y_0)-F(x_2,y_2)=F(x_0,y_0)-F(x_2,y_0)+F(x_2,y_0)-F(x_2,y_2)$ $\leq M_1(F(x_0,y_0)-F(x_1,y_0))+M_2(F(x_2,y_0)-F(x_2,y_1))$. Replacing $F(x_1,y_0)$ with $F(x_1,y_1)$ only increases the right side, and replacing both $x_2$'s on the far right side with $x_0$ only makes the right hand side larger by convexity. Finally, replace the last $x_0$ that we just added with $x_1$ without ncreasing the right hand side. This gives us $F(x_0,y_0)-F(x_2,y_2)\leq M(F(x_0,y_0)-F(x_1,y_1))$, where $M$ is $2\max(\frac{x_2-x_0}{x_1-x_0},\frac{y_2-y_0}{y_1-y_0})$. $M$ can also be the sum instead of the max, in which case we can drop the 2.

Edit: This argument doesn't work without additional assumptions; see the comments.

5 added 59 characters in body

You can use the triangle inequality to solve this by looking at each coordinate separately.

$F(x_0,y_0)-F(x_2,y_2)=F(x_0,y_0)-F(x_2,y_0)+F(x_2,y_0)-F(x_2,y_2)$ $\leq M_1(F(x_0,y_0)-F(x_1,y_0))+M_2(F(x_2,y_0)-F(x_2,y_1))$. Replacing $F(x_1,y_0)$ with $F(x_1,y_1)$ only increases the right side, and replacing both $x_2$'s on the far right side with $x_0$ only makes the right hand side larger by convexity. Finally, replace the last $x_0$ that we just added with $x_1$ without ncreasing the right hand side. This gives us $F(x_0,y_0)-F(x_2,y_2)\leq M(F(x_0,y_0)-F(x_1,y_1))$, where $M$ is $2\max(\frac{x_2-x_0}{x_1-x_0},\frac{y_2-y_0}{y_1-y_0})$. $M$ can also be the sum instead of the max, in which case we can drop the 2.

Edit: This argument doesn't work; see the comments.

4 Followed recommendation of qn

You can use the triangle inequality to solve this by looking at each coordinate separately.

$F(x_0,y_0)-F(x_2,y_2)=F(x_0,y_0)-F(x_2,y_0)+F(x_2,y_0)-F(x_2,y_2)$ $\leq M_1(F(x_0,y_0)-F(x_1,y_0))+M_2(F(x_2,y_0)-F(x_2,y_1))$. Replacing $F(x_1,y_0)$ with $F(x_0,y_1)$ F(x_1,y_1)$only increases the right side, and replacing both$x_2$'s on the far right side with$x_0$only makes the right hand side larger by convexity. Finally, replace the last$x_0$that we just added with$x_1$without ncreasing the right hand side. This gives us$F(x_0,y_0)-F(x_2,y_2)\leq M(F(x_0,y_0)-F(x_1,y_1))$, where$M$is$2\max(\frac{x_2-x_0}{x_1-x_0},\frac{y_2-y_0}{y_1-y_0})$.$M\$ can also be the sum instead of the max, in which case we can drop the 2.

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2 Minor error; added a 2
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