Linq query to get the distinct values in a list

Suppose this is my member class

class Member 
{
    public string CategoryId { get; set; }
    public string MemberName { get; set; }
    public int Distance { get; set; }
}

And, this is list.

var list = new List<Member>();
list.Add(new { CategoryId = "01", MemberName="andy" Distance=3});
list.Add(new { CategoryId = "02", MemberName="john" Distance=5});
list.Add(new { CategoryId = "01", MemberName="mathew" Distance=7});
list.Add(new { CategoryId = "03", MemberName="bakara" Distance=2});

Can anyone please suggest the logic/ linq query to get the List having distinct/unique categoryID with Shortest distance.

The output should be :

list.Add(new { CategoryId = "01", MemberName="andy" Distance=3});
list.Add(new { CategoryId = "02", MemberName="john" Distance=5});
list.Add(new { CategoryId = "03", MemberName="bakara" Distance=2});