2
$\begingroup$

I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space).

I was wondering whether the following (weaker) statement holds:

Let $X$ be a Banach space and suppose $A,B,C_0\subset X$ are bounded, closed, convex and non-empty subset. Do we then have $$A+C_0=B+C_0\implies A=B?$$

$\endgroup$
4
  • 1
    $\begingroup$ what is the difference with that question? $\endgroup$
    – Norbert
    Aug 14, 2012 at 20:03
  • $\begingroup$ There is an equality symbol instead of an inclusion, which makes this statement stronger $\endgroup$
    – gifty
    Aug 14, 2012 at 20:12
  • 2
    $\begingroup$ This statement is weaker, not stronger. Yours immediately follows from the other one (as shown in the answer) while the other one doesn't follow from yours. I find it objectionable to call the other question not properly answered, as the easy fix was presented in a comment while joriki's answer gave the important geometric intuition. $\endgroup$
    – t.b.
    Aug 14, 2012 at 20:34
  • 1
    $\begingroup$ I've now fixed my answer to the other question. $\endgroup$
    – joriki
    Aug 18, 2012 at 4:55

1 Answer 1

2
$\begingroup$

Modulo result presented in this question the solution is extremely simple $$ A+C_0=B+C_0\Longleftrightarrow (A+C_0\subset B+C_0)\wedge(B+C_0\subset A+C_0)\Longrightarrow $$ $$ (A\subset B)\wedge (B\subset A)\Longleftrightarrow A=B $$

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .