If a farmer's field was perfectly circular and had no corners, how would peah be designated? Would it be exempt from peah? Would he designate the outer ring as peah?
Strictly speaking, Peah doesn't have to be from the corners of the field. (Mishnah Peah 1:3). The mishnah says that the minimum measure of a field that should be reserved for peah is 1/60th (Mishnah Peah 1:2) although there's no maximum, so I would assume the same ratios would apply to a round field. Peah is also usually left to one side of a field, so it would have to be on one side of the circle (in other words, you couldn't leave it around the entire perimeter of a field, regardless of shape).
The Malbim on Vayikra 19:9 has an explanation of why the word peah is used in this context. He explains that the word peah can also refer to something which spreads out in four directions, even if it's not a regular polygon. I can't find a primary source for Malbim online, but there's this summary.
I will quote the mishnah and use logic on it then quote the Toseftah which seems to deal with what part of the field is to be left for Peah. I will then quote a Toseftah that seems to deal with how the loctation in the field is to be defined.
The first Mishnah in Peah says
The Peah has a minimum measurement, but does not have a maximum measurement [by Rabbinical decree]8 [If a person] makes his whole field into Peah, it is not [considered to be] Peah.
This would mean that the term "corners" does not have to be corners specifically. However, since most fields are rectangular, it uses the term to show the normal case. That is, the owner sets apart a portion of his field so that it is easily determined which parts the poor people "own" and which parts the workers are to harvest. An oddly shaped field should have a (relatively) compact and contiguous area or areas ("corners") for peah collection by the poor.
In the case of a perfectly circular field, I would say that the logic would imply a wedge or wedges (not necessarily going all the way to the center) at different areas of the field.
Tractate Peah, Chapter 1, Tosefta 5 actually says that a person can designate from the "beginning, middle, or end" of his field for Peah.
Tractate Peah, Chapter 1
A person gives Peah (corners of the field) from the beginning of the field,2 and in the middle [of the field], and in the end [of the field].3 But if he gave either [only] in the beginning [of the field], or [only] in the middle [of the field], or [only] in the end [of the field], he has fulfilled his obligation [of giving Peah to the poor].4 Rebbi Shimon says, “If he gave either [only] in the beginning [of the field], or [only] in the middle [of the field], or [only] in the end [of the field] it is considered to be Peah, but he [still] needs to give the proper amount5 [of Peah] in the end [of the field].”6 Rebbi Yehudah says, “If he left [at least] one stalk [in the end of the field,] he can add to it [from other parts of the field to make up the minimum amount and] it counts for him as Peah. But if [he did] not [leave even one stalk at the end of the field] he only gives [what he left in the beginning and the middle of the field [to the poor] as ownerless [produce, but not as Peah].” Rebbi Yehudah said, “When do we say this [that he can add the produce in the end of the field to the produce left in other parts of the field and all of it counts as Peah]? At the time that he [actually] gave Peah [by leaving at least one stock in the end of the field] and then he wants to add [to it more produce from other parts of the field].”7, 8
מסכת פאה פרק א
נותן אדם פיאה מתחילת השדה ובאמצע ובסוף. ואם נתן בין בתחילה בין באמצע בין בסוף יצא. רבי שמעון אומר אם נתן בין בתחילה בין באמצע בין בסוף הרי זו פיאה וצריך שיתן בסוף כשיעור. רבי יהודה אומר אם שייר קלח אחד סומך לו משום פיאה ואם לאו אין נותן אלא משום הפקר. אמר רבי יהודה במה דברים אמורים? בזמן שנתן פיאה ומבקש להוסיף
The Tosefta says explicitly that when it says beginning, middle and end it is referring to the locations of produce inside the field. However the Beraita in the Sifra (Kedoshim 1) says the same law as our Tosefta without the word “field” in it. Based on the text in the Sifra, Saul Lieberman in his commentary Tosefta Kepshuta (on this Tosefta), claims that some Rishonim (medieval authorities) explain that it is referring to the time of the harvest (i.e. beginning of the harvest, middle of the harvest and end of the harvest) and not to the physical location of the produce in the field. See Rashi (Shabbat 23a, Lesof Sadehu) and Rabeinu Hillel (Sifra, Kedoshim 1, Daf 40a, Veein Peah Ela Lebesof). However it seems to me that that is not the intent of these Rishonim, but rather they learn the Sifra in the same manner as this Tosefta as I will explain further. From the Tosefta it is clear that that is not the meaning of this law and that it is referring to the location of the left produce in the field, because the Tosefta says the word “field”. In fact, most other Rishonim learn it to mean exactly that. See Rambam and Rash Mishantz on Mishna Peah 1:3. According to this explanation it is a little difficult to understand what is meant by “the beginning of the field”, since fields do not really have a beginning and an end, but rather the center and the edges. Talmud Yerushalmi (Peah 1:3, Daf 6a) implies that these three locations are relative to where the person began harvesting his field and they do not refer to constant points in the field. So the beginning of the field means the place in the field where the person began harvesting the crops, the middle of the field refers to the spot where he has harvested half of the crops and the other half still remains, and the end of the field refers to the spot where the last of the crops have remained after the rest of the field has been already harvested. This explanation is preferred by Rash Sirillio in his commentary on the Yerushalmi. The Ralbag in his commentary on the Torah (Vayikra 19:9) explains this logic of the Yerushalmi as follows. He says that the Torah does not care where the person started and ended harvesting his field. He could have started harvesting it from a corner in a spiral circle and the last patch of produce that remained from the harvest ended up smack in the center of the field, which is the location of the field which is most difficult to access. Still the Torah prefers this last remaining produce to be Peah despite the difficulty of access to it. The Torah does not care how hard it is for the poor people to get to the left produce as long as they can get to it. All the Torah is concerned with is that the farmer leaves the last of his produce for the poor. The Torah did not want the farmer to feel that his top priority is taking care of the poor and not of himself; therefore he is only required to leave the last of his harvested produce and not the first of it. In fact Tosefta 1:7 points out four reasons why the Torah preferred that Peah should be left in end of the harvest path. Physical accessibility to the produce is not one of those reasons. Therefore Rashi and Rabeinu Hillel that are mentioned by Lieberman mention the beginning of the harvest not because they were talking about the time of the harvest season, but rather the location of the produce in the field where the farmer began harvesting it.