How do you get product management to buy into focusing the roadmap on a specific customer segment?

You need to understand which customer segments you will tackle first and in what order. Not only will this exercise help define your Go To Market strategy but it also will greatly impact your MVP. I like to think of this incremental Go To Market strategy as a customer acquisition roadmap. A product roadmap needs to align with a customer acquisition roadmap.  

So, you need to identify who your innovators and early adopters are and the customer personas/segments that fit within them in order to optimize your product at launch to meet their needs. Then, working with Product Management to align on those is the first step. Happy to dig into more details if you require here. Just let me know.

By presenting a particular customer segement in the light of the total addressable market. You'll be in good shape if you can clearly show a business case that carries signifigance in the following areas. For example, how does addressing a particular segment move the needle on revenue, competitive positioning, company vision and customer satisfaction vs addressing other segments? 

This is such a great question. It's not easy. The best way, which is not always (rarely?) possible, is to have enough knowledge and market intelligence to identify the ways two segments are different that it becomes evident that serving both isn't practical. For example, let's say your product solves a problem that both medium and enterprise-scale companies could benefit from, but you believe that medium-size companies are the right segment to focus on for now. What makes enterprise not worth going after? Do they have more complex intergration requirements? Security protocols? Customization requirements? If you can figure out the dimensions that make it challenging to serve both segments, you'll have a better shot at showing product that an everything-and-everyone strategic isn't focused enough to drive success.

If you have the right data and methodology, the value of the recommendation should be self-evident. There's a few steps to doing this well

1. How you segment: First to note is whether you're segmenting the market or your existing customers. Sometimes you'll want to talk about the market: our existing customers only represent 60% of the overall market and there's a segment making up 40% that we need to be going after! Other times you'll want to talk with product about a segment within your existing customers: these 10% of customers represent 80% of revenue, but everything we've built over the past 12 months has only been meeting the needs of the 90% low-value customers, so we need to better serve unmet needs of the 10% if we want to grow the business. Note that even if you want to have an existing customers conversation, you'll need at some point to have the market segmentation in order to increase your credibility that you're covering all your bases.
2. Build trust in the methodology: If you're going to product with a segmentation based on anecdotal evidence (e.g. you spoke to five sales reps and a couple customers), it is much more likely that your recommendation will be dismissed. Use a rigorous methodology that leaves as little to doubt as possible. Then make sure product understands that rigor. If they trust the methodology, they'll trust the numbers, and they'll trust the recommendation.
3. Use opportunity sizing to make your point: It's not enough to identify which segment(s) you think are most important to focus on. In order to put it in terms that will resonate -- not just for product, but leadership, too -- you need to get some $$$ impact estimates incorporated into your recommendation.

Happy to talk more about other aspects of this if you provide additional specifics of what your situation is!