It seems that mentioning bras gets people's dander up. So here goes. Why do woman wear bras?
1) Comfort
2) Modesty
3) To accentuate the breasts
Why should a man wear a bra? Usually the first two reasons. If you want to accentuate, it's fine by me.
It used to be that I only needed support for jogging, now I prefer it for walking and at work. I've seen those manly chest compressors and I'll be damned if I'm going to strap myself into something like that the rest of my life.
Sports bras seem a reasonable solution. There are basically two different types, compression and encapsulation. The compression type (which bear no relation to the men's compression garments) are usually simply constructed with no separate cups for the breasts. In contrast, the encapsulation type attempt to cup the breasts separately and are more similar to a standard "fashion" bra. For most men, a compression sports bra will decrease breast projection to about what you have when you lie on your back and at the same time prevent jiggling. The simplest sports bras are the no-hardware pullover type. To a first approximation they resemble a man's athletic undershirt, although they are shorter, i.e. a crop top.
A compression sports bra works to a B cup, possibly C. Local stores usually have sizes to 40, sometimes 42. Larger sizes will probably have to be ordered over the internet. I'd personally avoid anything with S-M-L sizing, especially if you're not brave enough to try it on in the store (I'm not). Just remember, stores are in the business of selling stuff. Just make your selection and go to the cashier, they can't shoot you. I find a plain white cotton Bestform racerback sports bra serves my needs. They go for about $4 at your local big box store.
And finally, if your breasts are actually that big, a regular softcup or underwire bra will be more comfortable than anything that compresses the breasts.
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