This question would probably be best asked on the crank length thread but since you asked it here I will answer it here.
The answer to this is not particularly simple. The simple answer is the athlete should choose the crank length that optimizes him or her. For climbing this is primarily a power consideration as aerodynamics plays little role, especially on steeper slopes and if the rider doesn't also have to go downhill also.
That having been said there are two considerations to generating power, force on the pedal and pedal speed. It is easier to put force on the pedals if they are moving slowly but high power also involves high pedal speed. Longer cranks have an advantage because they allow a higher pedal speed at any given cadence. Short cranks have an advantage because it is easier to apply force to the pedal because the knee has better leverage if it isn't bent so much.
So, in my opinion, it is all about gearing. If you find that 110 mm cranks is best for your power can you find a gear that will allow you to keep your cadence up to something reasonable for the highest slope you might encounter. This will be a lot different for a 6% slope than a 20% slope. Let's say you can climb at a sustained wattage of 250 watts. What gearing would you need? Well, analyticcycling.com says you would be riding at 4.92 m/s on a 6% slope and 1.66 m/s on a 20% slope. On 700c tires the wheel would be turning 2.24 rps at 4.92 m/s and 0.75 rps at 1.66 m/s.
Now, let's calculate the gearing gain necessary to not go below 60 rpm. 60 rpm is 1 rps. So, to keep rpm above 60 we would need a minimum "gain" of 1/2.24, or 0.446, at 4.92 m/s and 1/0.75, or 1.33, at 1.66 m/s. What does this mean for gearing choices? On the 6% slope the rear cog can be no smaller than 0.446 of the front chain ring on the 6% slope and the rear cog can be no smaller than 1.33 larger than the front chain ring.
Now, lets look at crank length. Lots of people think longer crank lengths give more leverage but this is wrong. The leverage is really dictated by the ratio of the crank length and the radius of the front chain ring. If the crank length and the chain ring have the same diameter the leverage advantage is zero because the ratio is 1. So 175 cranks and a 52 tooth chain ring provides a leverage advantage of 1.75/105 or 1.66 which means that the pedal speed is increased and pedal force is decreased by this amount for the same power and cadence. To get the identical leverage advantage on 145 mm cranks one would need a 43 tooth chain ring and on 110 cranks one would need a 33 tooth chain ring.
So, what does this mean? On 700C tires and generating 250 watts on a 6% slope and keeping cadence above 60 rpm one could ride that using a 52/23 combination on 175 cranks, 42/19 on 145 cranks and 34/15 on 110 cranks. So, for ordinary climbs (6% slope) it is possible to be easily fit your bike with the gearing necessary to climb. If you find that one crank length gives you a power advantage over the others then you should choose that one. However, when the slopes get very steep the situation changes. To maintain cadence the front chain ring must be smaller than the rear chain ring and the smaller the cranks the smaller the chain ring. Let's put a 36 tooth cog on back and see what is needed. So, to climb a 20% slope at a cadence or 60 at 250 watts requires a 27 tooth front chain ring on 175 cranks, a 22 tooth front chain ring on 145 cranks, and a 17 tooth chain ring on 110 cranks. Such a set-up is almost impossible to attain (although we have made a set of PowerCranks with a 23 tooth front chain ring for someone racing Mount Washington).
So, in conclusion, when the slopes get really steep, the only option is longer cranks. When the slopes are moderately steep shorter cranks are probably just fine, as long as you are properly geared properly for them, so then choose the crank that is best for you.
Hope my math was correct. I am sure some of you will correct me if I made a mistake. Either way, the correct math won't lie and this will give you the general considerations for making this choice.