# Does a convex set look the same near its face?

Recall that a face of a convex set $$X\subseteq\mathbb{R}^n$$ is a convex subset $$F$$ of $$X$$ such that every line segment with endpoints in $$X$$ whose relative interior meets $$F$$ lies entirely in $$F$$.

Let $$F$$ be a face of a (closed) convex set $$X\subseteq\mathbb{R}^n$$. Let $$x,y$$ be two points in the relative interior of $$F$$. Does there exist $$\epsilon>0$$ such that $$(X-x)\cap(\epsilon B)=(X-y)\cap(\epsilon B)$$, where $$B=\{\,z\in\mathbb{R}^n\mid \|z\|\le1\,\}$$ denotes the unit ball and $$X-x=\{\,z-x\mid z\in X\,\}$$ denotes the set $$X$$ translated by a vector $$x$$?

I guess it's true for polyhedra. Is it true for general (closed) convex sets, can you please point me to a proof or give a counter-example?

• You need to rethink what you mean when you say that "$X$ in the vicinity of $x$ and $y$ looks the same": taking an intersection with a ball $B$ centred on the origin isn't telling you anything about what $X$ looks like near $x$ or near $y$. (My earlier comment about equality stands, but this point is more important.) – Rob Arthan Apr 11 at 22:06
• How do you define the face of a general convex set? – David M. Apr 11 at 23:36
• I added more details to the question, incl. the definition of a face. – Tom Werner Apr 12 at 7:47
• Perhaps I am misunderstanding something- if you consider the closed square. The sides are 1-dimensional faces while the corners are zero dimensional faces. Locally the square cannot be the same at a point in a one dimensional face as it is at a zero dimensional face. Also your definition of "looking the same" doesn't account for rotation, so while I think you would like to say that the circle looks the same at any two points on the boundary, no two points on the boundary will satisfy your equality. – Eric Apr 12 at 14:09
• But the points $x,y$ lie in the relative interior of a single face of the square. As for your second question, I removed the (informal) sentence about 'looking the same' to prevent confusion. – Tom Werner Apr 12 at 15:46