# Quantified running

I am obsessed with numbers even more than I am obsessed with running. I do all kinds of calculations with the numbers recorded by my running watch. Given my training runs, how am I going to do on the next 24K? Should I target 12K where I could run better? So far, however, I have predicted my race times with mixed results: two hits, two far misses.

Why did I miss? I knew I had to push harder on races than on training runs, but my estimate of how much harder I can push came out of thin air. I went out too fast and lost steam. Here is my new system for race goals that I think will work better.

Suppose each run can be quantified by distance and time. Forget elevation, terrain, temperature, heart rate for now. In the graph below, each run is represented by a blue dot.

If my best pace is monotonically worsening in distance (a fair assumption), then the best time (the orange line) is a convex function of distance. This means that the set of feasible `(distance,time)` combinations I can do is convex. Believe me, I just proved this over coffee.

For example, if I ever ran 5K in 22 minutes and 21K in 104 minutes, I can run 10K in 48 minutes. Or 15K in 75 minutes. Or anything in between. In fact, I can probably run faster than that. Note how the orange line is always below the straight blue line.

More formally, my best performance can be characterized by the lower envelope of the convex hull of my runs in `(distance,time)` space. Anything on or above this line, I am capable of.

This insight lead me to the following conclusions.

1. Average performance doesn’t tell me much about race goals. Most of the time, I am not running anywhere near my potential, but that’s ok.
2. On races, I should target pushing the envelope a little bit. I can set a reasonable pace that I am sure I can sustain, and maybe even run faster. This way, I am setting myself up to succeed, not to fail.
3. I can pick races based on where I think the envelope is more likely to bulge. For example, if I do many near-envelope runs at 15K, I should do pretty well on a 15K race, even if I never did one before.

Here is the lower envelope of my actual runs form the past five months. Since I do some of my runs on the trail, I have adjusted for elevation. I plot it in `(pace,time)` space to make the patterns more visible, but everything I said above applies to this graph, too.