Given $A, B, C, D$ in $Oxyz$ space, find $M \in CD$ such that $MA + MB$ is smallest. Why can't I use AM-GM to solve this?

Question

In the $Oxyz$ space, consider four points $A(-1, 1, 6),$ $B(-3,-2,-4),$ $C(1,2,-1),$ $D(2,-2,0).$ Find $M \in CD$ such that $△MAB$ has the smallest perimeter.

As $AB$ is constant, the task is equivalent to finding $M$ such that $MA + MB$ is smallest.

$M\in CD$ so $M = (1 + t, 2- 4t, -1 + t).$ At this point, I could use calculus to find $t$ such that $MA + MB = \sqrt{f(t)} + \sqrt{g(t)}$ is smallest, and such $t = 0.5.$

However, I wanted to get the result faster so initially this is what I did:

$MA + MB \geq 2\sqrt{MA \cdot MB}$

So $MA + MB$ is smallest when $MA = MB \iff f(t) = g(t).$ However, this results in a different $M$ and a much greater sum of $MA + MB.$

I couldn't find what was wrong with this approach. Please help!

Answer 1

That kind of argument only works when

1. one of the 2 terms $(MA + MB)$ or $(MA \cdot MB)$ is constant (or somhow bounded in the right way), and
2. equlity of terms is possible.

Since 1) is not fullfilled, your argument is invalid. For any choice of $M$, say $M_1, M_2, \ldots$ you have

$$(M_1A + M_1B) \ge 2 \sqrt{M_1A \cdot M_1B},$$ $$(M_2A + M_2B) \ge 2 \sqrt{M_2A \cdot M_2B}$$

a.s.o. but since there is no relationship between $\sqrt{M_1A \cdot M_1B}$ and $\sqrt{M_2A \cdot M_2B}$, you cannot draw conclusions which sum is smallest.

You have taken a mental shortcut from the correct statement

"When two non-negative real numbers have a given positive product, their sum is smallest when they are equal."

to the incorrect statement

"When two non-negative real numbers are in some complicated relationship, their sum is smallest when they are equal."

In our case, the 'complicated relationship' is that the two numbers represent lengths of line segments, which is certainly not characterized by "their product" is constant.

Answer 2

It was shown in this answer that the locus of all points whose sum of distances from two points $A$ and $B$ is a constant $(2a)$, is an ellipsoid of revolution whose equation is

$ (p - C_e)^T ( a^2 I - U U^T ) (p - C_e) = a^2 (a^2 - c^2) $

where $p = [x,y,z]^T , U = \dfrac{1}{2} (A - B) $, and $ c = \| U \| $

and $C_e = \dfrac{1}{2} (A + B) $ is the center of the ellipsoid. At the minimum $a$ , the segment $CD$ passes through this ellipsoid at exactly one point.

Now, parametrically,

$ M(t) = C + t \ CD = C + t d $

Substitute $M(t)$ into the equation of the ellipsoid, you'll get an quadratic equation in $t$. Set its discriminant to zero and solve for $a$.

$ (C - C_e + t d )^T ( a^2 I - U U^T ) (C - C_e + t d) = a^2 (a^2 - c^2) $

Set $v = C - C_e $, then

$ (v + t d )^T ( a^2 I - U U^T ) (v + t d) = a^2 (a^2 - c^2) $

This is the quadratic equation

$ a_2 t^2 + a_1 t + a_0 = 0 $

where

$ a_2 = d^T (a^2 I - U U^T ) d $

$ a_1 = 2 d^T (a^2 I - U U^T ) v $

$ a_0 = v^T (a^2 I - U U^T ) v - a^2 (a^2 - c^2 ) $

Set the discriminant to zero, then

$ \Delta = a_1^2 - 4 a_2 a_0 = 0 $

i.e.

$ \bigg( a^2 d^T v - d^T {UU}^T v \bigg)^2 - \bigg( a^2 d^T d - d^T {UU}^T d \bigg) \bigg( a^2 v^T v - v^T {UU}^T v - a^2 (a^2 - c^2) \bigg) = 0 $

Now, let

$ w = a^2$

$ c_1 = d^T v , c_2 = d^T d , c_3 = v^T v , c_4 = d^T {UU}^T v , c_5 = d^T {UU}^T d , c_6 = v^T {UU}^T v $, then

$ ( c_1 w - c_4)^2 - ( c_2 w - c_5 )( c_3 w - c_6 - w (w - c^2) ) = 0 $

Expanding this cubic polynomial, we get

$ b_3 w^3 + b_2 w^2 + b_1 w + b_0 = 0 $

where

$ b_3 = c_2 $

$ b_2 = c_1^2 - c_2 ( c_3 + c^2 ) - c_5 $

$ b_1 = - 2 c_1 c_4 + c_2 c_6 + c_5 (c_3 + c^2) $

$ b_0 = c_4^2 - c_5 c_6 $

Note that $b_0$ is identically zero. Therefore, the cubic polynomial reduces to

$ w ( b_3 w^2 + b_2 w + b_1) = 0 $

Since $w \ne 0 $ (because $a$ cannot be zero), then our $w$ is one of the two roots of the quadratic

$ b_3 w^2 + b_2 w + b_1 = 0 $

To determine which of the two roots is the correct one, we need to keep in mind that $ w - c^2 = a^2 - c^2 $ must be positive.

I've applied the above formulas to the given points, and found that

$ a = 6.538573313 $

So that the minimum sum $MA + MB = 2 a = 13.07714663 $

To find the point $M$, recall that

$ M = C + t \ d $

The value of $t$ is determined from the quadratic

$a_2 t^2 + a_1 t + a_0 = 0 $

The value of $t$ at which this quadratic has its unique root is

$ t = - \dfrac{a_1}{2 a_2} $

Using the equations above for $a_2$ and $a_1$, one gets

$ t = \dfrac{1}{2} $

So that,

$ M = (1, 2, -1) + \dfrac{1}{2} (1, -4, 1) = ( \dfrac{3}{2} , 0 , 0 ) $